First thing to check is the oil pressure sensor. They do go bad and will give the exact result your speaking of. First thing would be to take out the sensor and rig in a standard pressure gauge. If your pressure is good then it's just the sensor. If it's bad then that's a whole other thread topic
Should be a canister looking device that threads into the back of the block or head. Usually a standard 1/8" or 1/4" MNPT thread.