If it turns out to be the core you may think about rebuilding the heater box while you have it out. The foam and seals go bad over time and the box leaks when they do. It's very simple and the system will operate more efficiently with new seals.
http://www.ebay.com/itm/DMT-MOPAR-6...m43a56b5a64:m:mkY6exhr7lP-S2vlT2gdc_g&vxp=mtr
Link to the foam kit.
Good luck.