What can cause my radio to transmit nothing but a terrible noise?
Being able to talk inside the airplane is not related to the radios function, the intercom (generally part of the audio panel these days) is what allows cross talk between headsets.
It sounds like you have checked all your connections which is always a good first step but after that:

Check your ammeter when the radio is in operation and make sure you have sufficient current to run the radio, if you are showing a heavy discharge on the system you may not have sufficient power to run the radio (the presence of 12V or 24V does not indicate you have enough power to run the radio)
Check your radio grounds (also lots of good info here) as they can be the source of a lot of static.
If both talk switches cause the issue (and they are wired properly) I would take a look at the radio unit its self. Perhaps have it pulled and bench tested with known good components to isolate any other factors from the aircraft. This may also assure you the radio was serviced properly when it was serviced.