I have a 1977 Cessna R172K. You've not said what your a/c is, but my experience seems similar and may give you a clue

My plane is a 1976 Cessna 172M (Skyhawk), and the Narco is a slide out
replacement for the Cessna radios.

I discovered a relay hanging under the instrument panel that is activated whenever I press the yoke-mounted transmit button. Among other things, it interrupts the receive side to the headset - so no sidetone when plugged directly into the jacks under the instrument panel

I have PTT on both yokes with intercomm jacks on both sides. The
problem seems to exist in both sides, although that doesn't mean they
both don't run to a common relay??? I can certainly look... ;-)

After concerted effort at troubleshooting, I discovered that one of my NAV/COM trays was actually 1/8 " longer than the other! Even with the radio fully seated, the plugs at the back were barely making contact. I found a used tray at a local avionics
shop that was the same length as the shorter one

This is seeming more and more like a likely possibility. I'm
considering taking the plane back to the avionics shop and seeing if
they can troubleshoot an issue between the tray connector and the
radio.

Thanks much for the response!
Todd