pull the sender unit out and turn car over, see if oil is coming out, if yes, then unit could be bad.
Pull wire and check for current, when car is running, otherwise the cost of a new sender is cheap enough(less than $10) for the warning light unit if you suspect it's bad.
The unit only comes on when there is no oil pressure at the sender or extremely low.

You could have a wire shorting out in the dash harness this seems to be what it sounds like. unless you replaced the wire harnesses with new ones, then you are using 45yr old wires like me, I've gotten use to the gremlins. My turn signals work one day then the next they won't work on the dash gauges only, but they'll work on the frt and rear of car. Mine could be an issue with the cluster? It's just not a issue I'm ready to deal with, I do plan on replacing all harnesses and having the gauges rebuilt, shes a project in the works.