Richard, I believe a model 09734-3 is a Primepact 66LP correct? I'm afraid I don't have any personal experience with that model. I'm only familiar with the Impact 36LP that's in my camper. There are a lot of similarities, though.
I CAN tell you that on mine, the PRIME switch does not really prime the engine with LP, it only opens the LP fuel lock-off solenoid, and that lock-off solenoid SHOULD be held open (by voltage) while the engine is running. There are many things that could cause the control board to shutdown the engine, and it does that by dropping voltage to the fuel lock-off solenoid.
The fuel lock-off solenoid allows LP to flow to the next component, which is a big round "pancake shaped device" that looks like an LP regulator. The maintenance manual calls it a secondary regulator, but it's not really a regulator (the LP fuel is already at the proper pressure at this point in the system) it's more of an on/off valve that only allows LP fuel to flow to the engine when there is vacuum in the intake manifold. Like when the engine is either cranking or running. It takes a different type of secondary regulator to be able to really prime your engine. But again, the lock-off solenoid should open every time you pres the prime switch, and stay open when the engine is running.
One thing that MIGHT be causing it to shutdown, and this is true on mine as well, is a bad oil pressure or oil high-temperature switch. Those are both located on the oil filter head (which is right out in front on your model), and a quick test you can do is to simply unplug them and see if the engine stays running without them. The wiring diagram for the 66LP shows those two switches wired in series (wire #85), and there should be a single-pin connector accessible that you can unplug to disable both of them. On my 36LP it's a purple wire.
Make sure you have sufficient oil in the engine first! The oil pressure switch is normally closed, and is held open by oil pressure, and the high-temperature switch is normally open and closes on high temperature. If either one is bad and is staying closed all the time, the engine will start, but will then shut down. That's actually one of the listed problems in the troubleshooting guide (section 8.1) for the 66LP. Possible causes are bad oil pressure or high temperature switch.
The Maintenance Manual I have for the 66LP is for a 09734-5, which is just a later model than the 09734-3 you are working on. There are probably a few physical differences, but for this problem it should be close enough. You may be able to find a -3 manual out there on the web somewhere.
I also sent you a PM.
:):)