I'm not familiar with the relay in the Cobra 29. Is the relay an "open frame" relay? That means there's no cover over the relay. If it is then it's possible the relay armature is dirty causing the relay to 'stick'. Changing microphones will not cure the problem.
If the relay is an open frame relay then you can clean the armature pivot point with contact cleaner and canned air.
Dust and dirt collects in and around the relay's armature pivot point preventing the armature from releasing, hence the relay appears to remain energized even though the PTT button has been released. Cleaning the relay's armature pivot point with contact cleaner and generous use of canned air can eliminate relay hang and allow the relay armature's spring to return the armature to its resting state.
Mike