Gas dude is here now. Just a weird interaction between the water heater venting and the furnace being on that overloads the exhaust vent because of the way the vent tees together, which is why it's intermittent. Only happens when the water heater and the furnace are going at the same time.
He told me how to fix it and is just finishing up. No crazy death traps in the house. Sorry to disappoint!