There are no oil passages in the LIM, but gaskets can fail, and coolant can get sucked into the cylinders, which usually does not end well, you can throw a connecting rod.
How many miles are on your car, if over 60k or so I can almost bet the LIM Gaskets are starting to fail around the coolant passages, so while this may not fix your smoking exhaust issue, it is something that I'd fix if I were planning to keep the car for a while.
I have not had to deal with an issue like you describe myself, but given the fact that heads are pretty reliable on the 3800 II, I doubt the smoke you get is from a head gasket problem, some of your piston rings could be damaged though, if you are able to I'd have the cylinders compression tested before investing in replacing the LIM gaskets, that way if you have to pull and rebuild, or replace the motor, you won't be throwing away the money for the LIM Gasket Kit .
If you do a compression test, let us know your results.
The lowest reading cylinder should not be less than 70% of the highest and no cylinder reading should be less than 689 kPa (100 psi). Perform compression test with engine at normal operating temperature, spark plugs removed and throttle wide open.
I'll go ahead and give you part numbers in case the compression test numbers are all good.
LIM Gasket is FEL-PRO Part # MS98014T
This kit comes with the reduced diameter EGR stove pipe for the LIM, this will give you added piece of mind that you won't have the EGR passage effect by the heat like it was before.
If you plug the coolant ports on the LIM to the UIM, then you do not need to worry about this.
If you do not plug them be sure to put the reduced diameter EGR stove pipe into the LIM.
The Dorman coolant elbows are fine to use, but not the o-rings on the old HELP pack, if you cannot get the the aluminum one, then you can either get the elbow from GM, or get the Dorman Nylon one, just be sure to replace the o-rings on there with these o-rings. ACDELCO Part # 1534066
The best fix for problems that are known with the elbow would be the aluminum elbow, it was back-ordered at a few places, but it looks like it is available now on RockAuto for about 7.00
Dorman 47065hp is the aluminum elbow part number, as far as I know the o-rings on the new aluminum elbows are not bad like the old ones were, you may want to pick up a pack of those AC Delco O-Rings just in case, they are pretty cheap.
Dorman Products - 47065hp