Well...
The shops told me that my Heater Core was clogged (98 Accord V6 sedan), and they were correct. However, instead of replacing it I took off the hoses that connect to the core at the firewall and attached equal diameter vinyl hoses. I then flushed the core with CLR several times, followed by hot water several times until there were no more sooty brown deposits coming out.
I drained the system prior to all of this- and also removed the radiator and soaked it in CLR overnight just in case. I also replaced the upper/ lower radiator hoses and the thermostat (the old one was working fine, but since I was already there...).
I now have heat in my car again for the first time in 20 months- and I'm only out a fraction of the cost to replace the core entirely. I feel that I did it right also, I took my time and did a thorough job. I don't feel that I cut any corners.
Hope I'm not posting old news, but I didn't find much on this by searching the forums and hope this will save someone else aggravation as I heard from the dealer that the 6th gen accords have this problem (they've done 10 or more in the last year.