I am a moron.
The reason the amount on the letter from the bank and the amount in the online mortgage calculator differed by about $35k is that the bank didn't take into account my down payment. They didn't look at my bank statements and determine how much I could put down (the way college financial aid apps do, which is my only previous experience with this sort of thing) -- they told me how much they'd lend me.
So. Thank you all for sharing my freak-out. I am sorry that I took up any of your day with it. (Though I really, really appreciate the support and encouragement and ideas, because you all know by now that that is how I deal with freak-outs.)
Big enormous thank you to
fairdice and
mayica, for pointing this out!
*hides face in shame*