Whats the mileage on the motor? If it has had a hard life, it could be that the valve train is clogged and not letting oil to the lifters.
I have a 400M in my 79 F150 and the valves are not adjustable I don't know about the 351. You may want to check that as well. Either way I would pop a valve cover and see how clean or nasty it is in there.
i just did a overhaul on the engine, thus i know that the push rods are not clogged up. yes the 351m is like the 400m in that you can not adjust the tappets, you can only get longer or shorter pushrods. but the pushrods are all within spec. so i am just not sure if the engine is leaking the oil pressure someplace else aka the crankshaft, or pressure valve, or gasket. or if there is someplace else it could be loosing pressure.
you have a guage in the dash of your truck. so what does it say when you start it, if the pressure is up and good, then most likely you have done soemthing wrong during the overhaul , my guess, soemthing is clogged somewheres, were the heads rebuilt? if not they could be clogged up and not be getting oiled, so yea start it up without a valve cover on and check whats going on, maybe you just have a loose rocker....never knoww