Cadillac Repair: Cadillac Seville 1999, intake manifold gasket, upper radiator hose

Question
I recently took my car to a repair shop (it was leaking coolant every 2-3 weeks there would be a pool of coolant). They replaced the reservoir tank, added an upper radiator hose, lower radiator hose. The car now does not leak huge amounts of coolant. However, I can't drive it very far (1-2 miles) and the "Check Coolant Level" lights up on the panel, and the engine begins to smoke. I returned to the shop, and they replaced the intake manifold gasket, gaskets for the cross over pipe and gaskets for the cross over pipe upper, more coolant and engine cleaner.
The car still has the same problem. When I called and asked if it "just might possibilty be" the thermostat, the mechanic paused and said... "yeah, it could be that". I am wondering. Does the smoking engine and the fact that the add coolant light goes on when I drive the car for 1 - 2 miles and the engine is clearly running hot. Does it sound like this might be the problem? And would they normally have checked out the thermostat with the original repair of the radiator leaking coolant? I'm in the SF bay area and I have already spent $1,700. on this. At least it isn't the 10,000 engine rebuild that was recommended by the Cadillac dealer for the leaking coolant issue. Thank you for your help.


Answer
I would replace the thermostat FIRST and if the engine continued to overheat without having any excessive external leaks from the radiator or the coolant recovery tank, I would have to say your looking at head gaskets that are allowing combustion chamber pressure into the coolant passages and the head gaskets would need to be replaced PER FACTORY SPECIFICATIONS and MODIFICATIONS.