I have changed these lines on both my 1997 S10 Blazer & a 2000 S15 Jimmy. Raise vehicle on jackstands. Then place a piece of wood on jack, move jack under oilpan, and jack up the engine 1 - 2 inches. This will give you the clearance you need to be able to pull the oil lines off of the engine block. You won't be able to pull the lines through if you don't do this, because of the fitting on the end of the hoses is too big. A vareity of rachet extensions will enable you to reach the bolts (two) on the engine block (reach from in front of the cross member to the rear). The lines between the remote oil filter & radiator ( cooler) are much easier to get to.
If it is a manual transmission you do not have transmission cooler lines. If its an automatic they will be on one side of the radiator and you can follow them to the tranny from there.
the cooler is part of the radiator.
Oil cooler line
Jimmy Ball died in 1988.
Jimmy McCambridge died in 1988.
Jimmy Martinez was born in 1988.
Jimmy Norris was born in 1988.
Jimmy McGuigan died in 1988.
Jimmy Goonan died in 1988.
Jimmy Launders died in 1988.
Jimmy Fidler died on August 9, 1988.
Must be transmission fluid leak somewhere, and it's getting fluid on the exhaust. I'd be looking at the lines to the cooler first.