I turned down the thermostat on the water heater and the pilot light stayed lit.
Probably the problem is in the thermostat safety circuit.
It is very low. I will try seeing if I can adjust it higher and have it still work.
That will at least buy me some time while I work out getting a new water heater.
I tested the shower head flow rate with a bucket. I have to find out how big the bucket is, I estimated 3 gallons. It took 1 min 20(ish) seconds to fill from the shower. 2.25 gallons per min.
(That is 3.75 if it was a 5 gallon bucket)
I need to look up the flow rates of our appliances.
That will help me select the size of heater we need. (Minimum for turn on and max flow)
I need to measure:
Vent size - needs 5 inch vent diameter
Gas line size - needs 3/4", not the typical 1/2"
May need a electrical outlet ran.