1998 Sport engine surge, also thump in back

I have a 1998 Explorer Sport, auto trans;

1. when engine is warmed up to running temp, at idle I hear a relay switch, then the engine powers down just as if the A/C has come on, then after 2 seconds the relay clicks again, and the engine is back to running normal, then relay clicks, and engine powers down....this cycling contiues for the longest time, at idle, sometimes I can leave it at idle for 1/2 and hour, and this "surging" cycle keeps happening. All controls in the truck cab are off, by the way.

1. When ever I brake to a full stop, just before coming to a standing stop, there is a loud CLUNK (which you can also feel in your back) coming from the rear. If you come to a more gradual stop, it doesn't seem to happen. Could this be a tranny problem, or something else ?

thanks