A big thank you to everyone who provided repair suggestions for my rig! Your suggestions were all very appreciated!!
I finally found the culprit--L404 was open so I soldered a wire around it. Here's a link to the repair if someone needs more details http://dr-fix-it.blogspot.com/2013/03/htx-10-mic-audio-repair.html
. Thanks to KJ4LTC for his suggestion on this (posted here http://forums.qrz.com/showthread.php?284488-HTX-10-mic-problem
Prior to a proper diagnosis I did use the "shotgun approach" and made several repairs hoping that one would do the trick: I re-heated all the solder joints as suggested by K1CJS and KZ1X, and replaced the mic element as suggested by W5WSS. Then again, one of these joints could have been bad as well, so all the work was well worth it.
The big delay for me was building a linear power supply so I could work on the radio. The computer power supply I was using would not maintain the proper voltage to properly diagnose the problem, so I purchased a automotive battery charger from Harbor Freight for the transformer and the diodes, and then added a LM317 regulator to it along with some filter caps and control transistors--now I have a regulated adjustable supply, and it only took a year!
Once again, thanks to everyone for all of your suggestions and assistance.