According to GM 1 quart per 1,500 miles is an acceptable oil consumption tolerance. 1 quart in 3k isn't bad.
I've had several gm 3.1 and 3.4 liters that have had there oil consumption issues remedied after the lower intake gaskets where replaced due to coolant leaking.
There are a couple of common problems that can cause oil consumption. Valve seals, bad rings, and PCV valves.
Valves seals would require a valve job which would requirethe removal of the heads. Rings would require a motor rebuild. Both of these are expensive options and oil is still pretty cheap.
Replacing the PCV valve might help. On the 3.8 it will be under the MAP sensor on the passenger side of the intake plenum.
As for as the intake gaskets, its a possibility but I don't have any hard evidence this could be the problem on a 3.8. Eventually the gaskets will start to leak oil. If you decide to replace them at that point I guess you will find out. I wouldn't do it on a hunch, its not a cheap repair.
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