5,500 watts is not a lot of power for a whole house. The Oil heater and air handler would use a bunch of that power, add in a refrigerator or two, plus lights, chargers and a bunch of other things and you can easily go over your wattage. You probably will eventually trip the circuit breaker on the generator, but maybe not. More likely your voltage will drop and things just won't work. You really don't give any detail as to your set up, but with this small of a generator, I would switch from the street power to the generator with all loads off, which I would probably do by shutting off all the circuit breakers, start the generator, then selectively turn on only the heat and maybe a light and tv. When the well needs to run, turn all off again, the turn on the well. There could be something else wrong, but that's where I'd start. I used a 5,000 watt generator in an old house, I used one of these switches to manage the load.
https://www.lowes.com/pd/Reliance-50...TCH/5015292865
It worked well. When I moved up here full time, I installed a 24kw propane generator with a 200 amp automatic transfer switch. This setup runs almost everything in the house and has worked very well over the past couple of years.