It is logical to think that if pump works while priming, then it is not 
the problem. However, there is a component in pump that fails when it gets hot (on certain pumps) Airtex does make this pump for Onan. You can get from NAPA.  Onan part # is A047N929.  From my experience as an ONAN tech (30+years) Your symptoms indicate bad pump, or ...hopefully not....control board.  Since pump is 1/4 cost of board, that would be first step.  I have had control boards, when they get hot, break down and quit supplying voltage to pump.