97 f-150 4.2 ltr - no oil pressure

It sounds like you have some internal engine wear. What causes the pressure to drop when you are at a stop is the engine RPM drops. When the RPM drops the oil pump does not turn as fast, when the oil pump doesn't turn as fast there is less oil volume, when there is wear in the bearings and even in the pump itself there is more volume of area that needs oil than there is available causing there to be pressure loss. The only fix for this is to replace the worn parts either by rebuilding your current engine or replacing it with a good salvage or new engine.

Three things, low on oil, low on pressure, bad sending unit. First one is easy, do you have enough oil in engine. Second, is the oil pump functioning properly. Third, is the pressure sending unit working properly. I recommend changing oil first if still occurs, replace sending unit, then if it still happens, i would have the pressure tested with a mechanical gauge. If pressure ok then drop oil pan and check screen. Screen clear replace pump unless there is alot of metal in pan then rebuild motor

When you had the pump replaced did anyone look at the rod and main bearings? 99% of all low oil pressure problems I've seen has been directly related to bearing wear, not a bad oil pump.
Replacing the pump will give you slightly better pressure temporarily but that won't last very long.
Verify the actual oil pressure using an external mechanical gauge. If it's low (likely it is) go back into the oil pan and use plasti-gage on the bearings to measure clearances.
In reality though, in an engine with that many miles, it's best to either completely rebuild or replace it though.

Why do you want to replace the oil pump? I am asking this question for good reason. Engines that have worn bearings will have low oil pressure, and replacing the pump will not solve this problem. If you are replacing the pump because of low pressure, then you really need to consider correcting the cause of the low pressure. To replace the pump, you need to remove the oil pan from the bottom of the engine, which will likely involve raising the engine. Once the oil pan is off, you can readily remove and replace the oil pump.

yes you will

Probably an oil pump, will it still do this if you rev the engine a little while stopped? If not then you have an oil pump that could be failing or hopefully it is the pressure sensor that is off. either way don't chance it

The best way to prime the pump hoping that you havent ran the engine to long without oil pressure
is to remove the oil gauge switch ORoil light switch on the engine and get a connector with the correct thread to screw into the block and fit a hose to it and to a 1/2 ltr oil can pump oil in as hard as poss and then turn engine over IN REVERSE(Backwards) 3 times and then (pumping every time) refitswitch and start it should be ok....

oil pump going, or the crank bearings worn.
are there any noises that appear when oil pressure drops?

check sensor

The oil pump is in the oil pan and needs to be dropped and being a 4WD you also need to drop the front axle pumpkin to remove the oil pan and replace all the gasket needed. Start with the easy and replace the oil pressure sending unit, it's on the back of the engine to the passenger side of the distributor. You can go to Auto Zone and pick up a new Oil sending unit, just make sure your get the right one, "Sending Unit" Looks the same are the oil pressure sensor but the unit reads the pressure, the other just signal a light to go on and off. It's an easy fix like replacing a spark plug. 

yes, the diaphragm in the fuel pump has a small hole allowing fuel to seep into the crankcase, thus diluting the fuel and giving you low oil pressure...Change the pump and change the oil and filter.