After the engine runes for 5 minutes the oil pressure fails and the check guages light comes on
engine loses oil pressure on 1998 Dodge Durango
ACME AUTO, ok. IF FULL OF GOOD CLEAN OIL!! - NAPA part # ECH OP6189 This is the oil sending unit or oil pressure switch, replace it! Next time tell us what size engine we are working on!!! However in this case i think this should cover it. This part should be on rear of engine passenger/right side of the distributor. Remove the little red locking bar to remove the connector, 1 1/16 six point deep well socket will fit it. I used snap-on as others may not go over the unit. Made easier by marking then removing the plug wires going to the right bank. Hell i might as well finish this, if that little red locking tab wont come out just reach back there with your hand and break the top off of the switch and pull it out where you can disconnect the harness. This should tell you how easy it would be to f... up the new one, just use some common sense.
have a manual oil pressure test done to verify its not just the gauge
After the other 2 suggestions , my 2cents worth would be to see if you can get a view of the oil pick-up screen next time you drain the oil. If it's a high mileage engine with only moderate to poor maint. , it's possible the screen is getting clogged with debris seen it a lot -same symptoms you describe.