Buy a new heater core, remove the dash, then open up the HVAC box and use paper towels or similar to soak up the antifreeze and then install the new heater core.
http://www.ramforum.com/f76/how_remove_dash_access_hvac_box-19058/
Removing the dash the first time is annoying, but not all that hard. Opening the HVAC box is a pain in the ass though.
https://www.youtube.com/watch?v=XCbGJ-bIkmg
I used the video above to open the box without evacuating the AC system. The bolts are super annoying and very hard to access. You'll understand why a heater core is like a $900 job at the dealership. But I would recommend doing it yourself because you learn a lot.
So in general summary:
1. Pull off dash trim pieces
2. Drop steering wheel
3. Disconnect electrical connectors
4. Remove dashboard
5. Disconnect and remove ECU (3 bolts are behind it, thanks dodge engineers)
6. Unscrew 6 or so bolts in the firewall to gain acces/wiggle room to HVAC bolts on the back
7. Open up HVAC box
8. Clean box and replace heater core
9. Put everything back together.
If your HVAC doors don't work i'd recommend replacing them as well since youll have access. You'll probably have to break or saw off several of the screws holding the HVAC box together behind it, as I have no idea how you could ever get access to them. Good luck, its a pain in the ass job but worth doing. Give yourself 2 days to do the job to be safe.