Have to ask, is this radio new, or recently bought used?
Several threads may help apply here, it's more reading but they cover a lot about the Cobra 29 - just not exact...
cobra 29 ltd no modulation output.
And also here...
Cobra 29 ltd classic not transmit
New radio? Check the radio for bad or cold soldering...even broken wire or loose case to board screws. Mic jack to chassis - loose hardware can make this a headache...
Used radio?
Several things can cause problems...
Some older radios - if someone installed an Echo board and then took it out might have goofed up the wiring for the Mic.
Capacitors - for an older radio like the 29 with ST - can fail - so you'll need to look at that too.
Mic or Mic's wiring - even plunger - can affect the "Chirp" you hear...
But, firstly I need to know, did you got this radio used, do you know it history...?
Check this on an external speaker with PA on - use the PA speaker jack , does it chirp in this mode?
Because it can also mean the rear panel jacks may have got issues.
Just some things I'd check...