Oil pressure issues in my 1992 Range Rover

Hi, I have a 1992 Range Rover with a 3.9. This is my first Land Rover by the way. I am having trouble with the oil pressure. When it is cold (Phoenix Arizona cold 40-50 degrees) I can start it up and the oil light will flicker for about 3 seconds and then go off. When I drive it 2 or more miles, up to operating temperature and cut it off and let it set for a minute or so and start it back up the oil light will come on and stay on. I loose oil pressure, I know this because the rockers will start making noise and I have checked it with an oil pressure gauge. I have dropped the oil pan, cleaned the pickup tube and screen, replaced both oil pump gears, inspected and cleaned the oil pressure relief valve and changed to Valvoline Racing 20-50 oil. I have been working on American cars for 20+ years and have never dealt with this type of issue. What I mean by this is as long as I drive the RR and do not turn it off from the initial morning start it will run great for as long as I drive it. So I have ruled out clogged oil ports as the oil is circulating as long as a do not turn it off, but as soon as I do it seems to not hold its prime. I can leave it sitting after it looses oil pressure over night and the next morning it will start up with the light on but as soon as the oil pumps primes the light will go off with no issues until it heats back up. Sorry for the long post but i tried to give as much info as possible. Thanks in advance for any advice.