For those of you coming across this thread later -
So your "check oil level" lights comes on.
First thing to do is check the oil level.
If you bought a used car, make sure it actually has the correct dipstick. Dipsticks are not all the same, even on different models of cars with the same engine due to differences in oil pans and such.
If your oil level is OK you may have a mechanical problem but most likely it's the oil sensor. I can't recall any postings about oil pumps failing on a 3800 NA or SC engine, although I suppose it's happened to somebody. Bad sending units are common, so many people will replace the sender without checking anything else.This approach might save time, but it's risky because unless you measure pressure directly with a gauge attached to the engine, you have no way of knowing if pressure is within specifications or not. SO - if the check oil level light is on and the oil level is OK, it's a really good idea to check oil pressure with a gauge.
One other thing - If your car did not get regular maintenance before you got it there may be sludge built up in the oil passages. Then what will happen is the oil pump will move the oil up to top of engine but it won't drain back down as it is supposed to. One way to guess at this is to open the oil filler and look in with a flashlight. How clean is all that metal?
Hope this helps.