I had this happen to me on my '97 supercab 4.6....definitely the heater core.
The dealer wanted around $1100.00 to replace the $37.00 core! 99% labor for this repair. I decided to tackle it myself with a Haynes manual at my side. Took 2 full days - you have to remove the entire dash to get at the air box that houses the heater core. A huge PIA......I paid the grand to have it done on my wifes '98 Expy when the hot/cold mix gate failed iin the airbox. I didn't need the back pain again.. I'm 6'3". I hope there are some other threads on this site, because it is a big job to take on. Good luck!