His life hasn't changed much thanks to his empathy. He's still been on the same path, still ends up on the ship, still holds the same rank, same job, still ex-Section 31. Mostly it exacerbates his already tenuous relationships with other people. It might not even be readily noticeable, as the ability does a good job of explaining why he's so closed off with people and yet also intensely emotional. (Of course, you can also understand this through normal canon, but that's not the point.)
He's had it since he was born and didn't know until much older that it was something special, something different, something unique (as far as he knows) to him. He worked hard to gain the approval of the father he knew didn't love him or the rest of his family, and the meek mother who was always anxious about her husband and about public appearance. His refuge was his older sister who was full of joy and love and was so much calmer and warmer than most of his hectic life. He was very bright but always very quiet in school and around classmates. As everyone got older, he always felt isolated, because he always felt such disinterest, or worse, hate from classmates. Still, he tried to make friends, and he had strong issues with some of the bullies.
Can you see where I'm going with this? Yes, the drowning still happened, and honestly, that may have been the most peace he'd had in his life outside of sleep. His father's attitudes changed over the months after, and the narcissistic pride turned slowly but surely into anger. This, in turn, would get Malcolm angry, even if he didn't intend to be, and things only escalated from there.
Reflecting the emotions of others had given him a dual, jarred sense of self. How could he trust his own emotions when he was constantly reading if not actually feeling those of everyone around him? Sure, as he grew, it became much easier to not feel them, only sense, but strong feelings and outbursts still messed with his head. This naturally made all of his teenage years extremely difficult, trying to find who he was in the midst of emotional, hormonal outbursts in the drama of middle and high school, or worse yet, the people who had legitimate mental/social/behavioral/emotional disorders. He had to eat somewhere quiet and lonely, frequently the library, and paying attention in class became quite the chore (probably another reason why he did so well, because he made a point to concentrate so hard). Simply walking the halls between classes was a nightmare he tried to ignore and keep inside so as to not lash out without reason like a freak.
Certainly there was also happiness, joy, positive emotion mixed into it, but think back to your high school years. Most of it was fleeting and didn't last very long, perhaps for the rest of the day if he tried to hang onto it or search it out, or just until he passed someone brooding about a bad test or a bad break up or being bullied. Of course, positive thoughts regarding him were always welcomed, however awkwardly. Especially by girls. Who quite frankly terrified him most of the time, but still. He was safest alone, or failing that, around friends having a good time.
It was in his later teens and through his 20's that he really started experimenting with how not to feel. Some tactics worked, some didn't, and it really wasn't until he'd gotten partway through his Starfleet training to realize that he had a legitimate ability and wasn't just empathetic in a normal sense.
Getting to that point was difficult enough. The day after graduating school, he packed up his things, took all the money he could from his savings, and headed off for Starfleet. He'd start his college education there in San Francisco, and someday, he'd be among the stars. (What could be quieter than that?) This, however, brought on a few new challenges, as he was no longer among the comrades he'd grown up with in his private school, community, and family. Complete strangers were just as unpredictable, and the big city was hard to deal with. But he coped however he could, through concentration, or singing songs in his head, or going over phrases or equations, just trying to put up a block. Eventually, after some prodding and confiding in a teacher who he had come to trust, it was revealed that he was, in fact, an empath, with a 'gift'. (He would come to loathe this word.) It wasn't explained to him how this was known, if this was just a bit of folklore and superstition or if this was an actual ability--and if it was, if there were others like him, with this or anything else. It was all kept very quiet and close to the chest, but it made him feel a little better.
Not terribly long after, when he was an eager young ensign, Harris of the group that would become known as Section 31 approached him with an offer he couldn't refuse. Protecting Earth's interest, getting out into space on exciting work, made him feel like an action hero. He was taught better how to use his ability, to actively scan specific people, to dig deeper under layers, and, most importantly to him, to protect himself with what became known as a psychic shield, though he didn't care for that name, either. Harris made good use of him, as a spy who could sense when someone wasn't being honest, when they were uneasy--usually when they were lying--was quite valuable in that line of work. And though he gained some very valuable skills, the longer he was at it, the more he questioned the morality of it all. It had all seemed quite honourable at the time, and at the start, but by the time he left, he felt as confused and conflicted as ever.
But he endeavoured to become a new man. Worked up the ranks, making use of his other, less fantastical abilities with weapons systems, explosives, fighting techniques, and the like. He'd become a cold, collected person who was all about duty and discipline, distant, hard to get to know. He didn't trust people emotionally. He knew he could trust shipmates, a captain, on duty, but in personal situations, he didn't know how to react and tended to shut down conversations before they began, withdrawing in on himself.
Once on the Enterprise, the same happened all over again. But the longer he was there, the more life-or-death scenarios he was placed in, the more people tried to know him, he started to slowly open up. Quick scans, now and then, especially when he wasn't sure how someone was feeling. And while he was still duty first with most of the crew, by the end of his fourth year on the ship, he had made some very, very close, personal friends with some of the members.
A FEW NOTES:
Some of Malcolm's canonical reactions and emotional states can be derived from his experiences. He at times can be quite moody, depressed, with a suicidal death wish of sorts. That stems from his near-death by drowning, when all was quiet and peaceful. He does, at times, feel that this can be easier, preferable to his day to day stressful state, but only at times. Likewise, his difficulty in social situations, in emotional situations, around people, as well as his own struggle with his own emotions, can all be explained in part by his life as an empath.
There are an extremely limited number of people who know, both in terms of people he's told and people who know despite that number. Mostly abilities are rare and super-uber-top secret like it's known secretly but not publicly I'll figure it out later so only like the top of the top would know, plus his confidant, plus Harris. That's basically it. By the end of season four, the other people who know are probably Trip (best friend, closest ally, possible interest in other ways), Archer (having to explain all the Section 31 bullshit in whole), and T'Pol probably has figured it out without him telling her (Vulcan + emotions + vague psychic readings). That's already way too many people to his mind, because he's extremely protective and paranoid about it.
Despite his protection, strong emotions and unexpected outbursts will still make it to him almost as strongly as if he didn't have a shield at all, depending on just how intense they are. That said, he keeps his shield up all. the. time. Unless he feels he has to feel someone out. Such times are when he suspects someone, or when he's trying to be comfortable around them, or just whenever he feels he has to. Strange people creepering about, you know, that sort of thing. He does this when he wants to know if he can trust someone.
He has not gone through a bajillion random hookup girlfriends. He has had sex a grand total of one time, and it was utterly overwhelming and pretty damn terrifying because of that. Getting physically involved requires an insane amount of trust and openness that's hard to come to terms with. And while canonical Malcolm couldn't get attached to these many girlfriends but always remembered their appearances, empath Malcolm places an emphasis on an emotional connection rather than looks. Which, given how he acts and how defensive he is with his ability, is really hard to do.
He started making decisions more for himself after his near-drowning, deciding after a few years to go to Starfleet instead of the Royal Navy. But it wasn't until he had the ability to really start blocking people out for longer periods of time, with or without the Section training, that he began to start trying to come out into his own, with his own emotions, his own convictions, his own thoughts. As such, even in his early 30's, he still struggles to find and ultimately understand himself.