I just replaced the heater core in my 2002 1500 quad cab (dual zone control). I had the same problems, no heat on drivers side, warm on pass. side. My core ended up corroding and leaking out of the back side. As it turns out, the rear of the heater core was plugged and that's the side of the core that heats the driver because the dual zone system uses two independent blend doors that are split with one set above and another set below the heater core inside the HVAC box.
I suppose flushing the core might be something to try, but I knew mine had been leaking for a while. The antifreeze even soaked and degraded the foam on some of the blend and mode doors bad enough that I had to replace it with some weatherstripping from the hardware store.
Sounds like some guys are having troubles with broken blend doors and such and many of them recommend the heater treater, but everything on mine looked good, besides the core and some rotten foam.
I decided to put in a new A/C evaporator too, since it was all in there together.
I'm really glad I did it on my own. Learned a lot through the process and saved a ton of $$.
Hope this helps anyone out there with similar problems. GOOD LUCK!!