Water on floorboards, help!

Everytime when I put my Airconditioning on or my Heat, the back floor boards get SOAKING wet with water. The passenger side does not get as wet as the driver side does. The front floorboards are DRY. I checked the coolant level and its filled up. This happened 2 weeks ago I thought I might of left the window down in the rain. But when its even dry out its still wet, and I checked the undercarriage there looks to be no holes in the bottom floorboard. The temperature gage stays in the middle so it looks like its not overheating. Its just the rear floorboards, the frontfloorboards and directly under the seats are DRY. The water is obviously coming from the car, how much water is in the car? I mean I put 4 full newspapers down and they get soaked, I put my hand in the rug and the water comes above my fingers!

Could this have something to do with the Heater Core? If so how much will I be looking at to spend?
Right now my money situation is tight, you think it would be alright if I just dont use the Heat or AC for awhile?

Thanks

BTW its a 2001 Lincoln Towncar Executive 78,000miles 4.0 v8