First off I would NEVER be afraid of oil pressure unless the lifters collapse! Not in a grocery getter anyway..
The spring in question is the pressure release valve inside the oil pump. It has happened that when the oil is cold and the pressure rises to beyond the spring pressure the valve has opened and STUCK OPEN. Thereby allowing a discharge of oil to KEEP the pressure from blowing out the bearings.
Then when the oil gets hotter and thinner the valve shuts and the pressure doesn't get low from the open valve... If this valve is STUCK OPEN low or VERY low pressure can be observed.
I would first do anything I could think of to assure myself exactly what the pressure is when it reads zero!! If, for sure, the pressure is really at or near zero I would take out the oil pump and look closely at the valve.. If it is open and I can repair it I'd do that.. Yep I am that kind of guy! But in your case I would prolly advise another oil pump.
Back in the day we used to take them valves apart and install stronger or weaker springs.. Most are simply a metal cylinder in a port (hole) with a spring on top held in by a friction pin. The way they work is simple. The pressure from the oil pushes the cylinder (valve) up against the spring.. As the cylinder goes up another hole is uncovered and the oil bypasses OUT of the pump and into the pan. By placing a stronger spring.. (or in some cases streaching the existing spring) the pressure goes UP until it can push the valve to the desired height.
It is not unknown that the springs can break, bind or otherwise fail.
Big Jim <img src="/forums/images/graemlins/pfft.gif" alt="" />