Problem solved. I contacted Jawbone support and they suggested a reset for my ERA and Icon earpieces. This did not work and I still was loosing the microphone on the earpiece when talking. The other party could only hear me when I talked directly into the RAZR. Also what was happening (since update 2.3.6) was that if I tried to use the phone speakerphone, after a minute talking with a party, the speakerphone would stop working and I would have to talk directly into the phone mike for the party to hear me. The bluetooth and speakerphone issues are the only two bugs I have experienced since the update.
Both of these problems were solved after talking with Verizon support. The tech person had me do a "Factory Data Reset". You can find this under "Privacy Settings" > "Personal Data". Keep in mind that when you do this you will loose data not saved including photos. All 3rd party apps will go, but will be updated back onto your phone from your gmail account. Unless you have done this before it would be best to have Verizon walk you through the process. After the Factory Data Reset, both of my Jawbone earpieces work and so does the RAZR's speakerphone.
This is great news, and certainly helps. I have also reset (Droid RAZR Utility, Fastboot .173), and haven't experienced the problems since either, but I wasn't ready to call it "solved" yet (which is why I haven't reported such), however with your similar results I would be more confident in that assessement.
Now, the question is what was similar or identical in yours and my user profiles that may have been the cause. To explain, I tried Safe Mode on my phone but didn't see any difference in the issue...that is, it still had "quirks", sound through both phone and earpiece, dropped sound from earpiece and no navigation sounds through earpiece or phone, yada, yada... Safe Mode is "supposed" to give you a pure factory software experience which would then point to a user-installed application. However the fact it happened for me while in Safe Mode points to a possible OS problem, or possibly something that changes in the OS over time.