GM SPECIFICALLY warns against using 10W-40 in most 3.8L/3800 V-6 engines.
I've used Pennzoil in GM, Ford and AMC engines for over 35 years and I've never experienced the problem you describe.
The more I think about this the more I'm convinced that the problem just may be the pressure relief valve hung open (or nearly so) by either a piece of debris or possibly the spring just gave up the ghost.
Before I dug into the engine any further I'd have the oil filter adaptor removed, pull out the PRV and the spring, then check the valve bore for any crap stuck in it. Also check the spring and the valve itself (a piece of debris stuck to the valve itself or a metal burr on the valve could make it hang in the bore).
If the pressure relief valve checks out OK, then you're either looking at a blocked pickup screen or another blockage on the suction side of the oil pump, a clearance problem on the pressure side (main, cam or rod bearings) that is leaking off all the oild pressure, or just a worn out oil pump (which is contained within the front cover).
Regards, Bill Bowen Sacramento, CA