Cadillac Repair: 99 Cadillac Deville, radiator cap, head gaskets

Question
I am having an overheating issue with the Cadillac that I just bought. I have flushed and replaced all fluids. It still overheated. Then, my mechanic noticed that the thermostat was not installed...someone took it out? We put in a new one and the car is still overheating every time I drive it. The fans are both coming on and at idle the temp holds steady, but as soon as I drive more than 10 miles it overheats again. The radiator doesn't leak...not sure what to do next and am almost out of cash. Any suggestions? Are there checks that I can do to see if the radiator or water pump is bad? My mechanic has given up. Oh, also, I had the oil changed but the digital readout is still saying that I need an oil change. Thanks!!!

Answer
This engine is well known for bad head gaskets which is why they took out the thermostat so as to try and get more coolant flow. You need to have constant air pressure put into each engine cylinder by removing all the spark plugs and watch for bubbles in the radiator with the radiator cap off.