LLoyd,
How it shuts down and runs should provide some clues. If it starts and runs normally and will take a load well and then suddenly just shuts down like you pressed the off switch then I would look to one of the sensors (like oil or temp) telling the unit to shut down, probably falsely. If the sensor systems are causing the shut down they use the fuel solenoid and you could attach a meter to those wires and see if the solenoid voltage is causing the shutdown.
If it misses or spits and sputters for awhile as part of the shutdown, it could be the fuel regulator. Propane sometimes has had excess oil in it that will settle in the regulator and interfere with the diaphragm and spring. This system brings in liquid propane so the stove which brings in just the vapor will not see the oil and will run fine. The heavy oil travels in the liquid stream. That oil can build up over the years. The restart, sputtering, and quitting leans toward the fuel system, possibly the regulator. If that is the case, with the fuel off, you would unscrew the regulator top and clean the diaphragm, housing, and spring of excess oil. Just some educated guesses from owning one of the LP Onans for a few years. The other simple thing to check is that you have no mouse nest in the air intake restricting air.
Later Ed