What year is this engine? If it's the older style pump on the side of the engine, those have to be packed with vaseline and well packed at that. If it's the gerotor style (pump around the crank), those *normally don't need any priming or packing* although the manual does recommend it. They are far less prone to losing prime or cavitation.
You could remove the crank pulley and remove the pump cover and pack it. That's always fun.
Gerotors rarely go bad and when they do you can hear it grinding badly (unless it's stripped). The older style (meshed gears) can get wear in the gears and the face of the pump but they will usually give some pressure (like 10 psi at idle but stop at about 25-30 PSI.
Stuck regulator valve will usually still give some pressure although looking at the rockers maybe not show much.
Neither pumps are easy to test since they aren't driven by the distributor.
Are you running an external oil filter? If so, switch the lines in case you have the pressure side to the inside of the filter thus forcing the bypass flaps closed. I did that once. No pressure. Reversed the lines and tada!
I don't know if any 3800's were produced without the gerotor pump. I have to assume that is what you have. I think gerotors was one of the changes for the 3800.
You might have to breakdown and pull the pan and check the pickup tube and pull the crank pulley and check the rotor gears.
I've have and had many Buick engines and other that having to pack the older pumps when the gears were removed, I've never had one go bad.
Whatever you do, don't allow the engine to start until you get this fixed. Buicks do not like running with no oil pressure. Trust me. You can crank it with the ignition disconnected until pressure comes up, just don't let it run. You should get pressure within 2 or 3 10 second cranks close together.
Don't get bummed. Work the problem. You'll find it.
[This message has been edited by TK (edited 08-26-2003).]