Leak onto floor passenger side from neck of ac drain through firewall

It's not your heater core. If your heater core is out it would leak antifreeze. My wife's 97 Grand Cherokee had the same problem. It is condensation from your AC coming leaking through a round hole in the AC unit at the firewall. Pull the carpet back and dry out underneath it, there will be a lot of water and we had to leave the carpet pulled back with a fan blowing on it all weekend to dry out. I fixed this about 8 years ago so if I remember correctly, the was a plastic part of the AC at the firewall that had a hole in it that the water was coming out of, I could see leaves and debris in the hole so I know it was part of the drain system. I cleaned out what debris I could and used a cone shaped peice of rubber and twisted it in to plug the hole and tapped over the plug to ensure that it wouldn't back out with vibrations, etc. After you dry everything off start the engine and turn the AC on max, you'll be able to see the where the water is leaking from after the AC gets cold. Hope this helps, it drove us nuts for a year before we figured it out and the wet carpet smell wasn't pleasant.

It sounds like your heater core is busted. you will have to go under the dash to get to it and replace with a new one