1999 Mercury Marquis Something with the Radiator?

Engine Cooling problem
1999 Mercury Marquis Two Wheel Drive Automatic 247103 miles

At first, it just vibrated and was very audible when turned off (loud, low rumble - not audible when driving). Then it overheated. I got it cooled down and put in more antifreeze. When I turned it on, it started to make asputtering noise that got worse when I accelerate. I stopped and went into a store, and when I came back out and turned it on, the noise was all but gone, but it would heat up then cool down, then heat up again. I've checked the Antifreeze tank and oil, and both look normal.