LTE version on Vzw, unlock/root, 4.0.4 IMM76K
Phone is from day of release on Vzw.
I have the exact same symptoms that the OP described, and this just started happening in the last week or so. Nothing has changed in my ROM/Kernel/Radios in that time.
I have installed the Wired Headset Routing app/widget as well, and I can tell that it's "trying" to override the phone's desire to route sound to the headphone jack, and sometimes it succeeds, sometimes it doesn't.
The best example of it is if I watch a video, say on YouTube. With no headphones plugged in, the video will start playing, either with or without any sound. Then, at some random point, it will pause the video, and if I hit Play to start it again, the opposite sound condition exists (e.g., speaker now vs nothing before).
It will then pause again, and when I hit play again, the sound routing has switched.
However, if I plug a set of headphones into the jack during this same video playback, the sound stays locked in on the headphones, and the video doesn't randomly pause.
This is very frustrating, as the same symptoms occur during phone calls, which makes my SmartPHONE basically useless as a phone.
I guess I should call Vzw looking for a hardware replacement, as the apps don't solve my problem.
Any other thoughts? OP, did you have any luck?