You have two oil pressure sensors. It sounds like the lower setting sensor (only goes off under 2000 RPM and no audible alarm) has gone bad. You can get them online for about $5.00 including shipping. Determine which one you have. It should be on the side of the cylinder head that the water hose connects to and will have a wire going to it. Note the color of the plastic when you remove it. The pressure setting is stamped on the bottom and can only be seen if the sensor is removed. Get the setpoint and plastic color before you try to order.
There is a chance your engine may be getting worn out and it's not maintaining adequate pressure when the oil is hot. Higher viscosity oil can help for while or Lucas oil stabilizer. I expect the problem is with the sensor though.