Near as I can tell, no car manufacturer repairs remote fobs beyond changing a battery.

E bay vendors are selling single button remote housings ( no internals ) . There are two versions, narrow and wide key but you should be able to reuse your key bit.

When turning the key, hold it close to the ignition lock, this reduces pressure on the housing and keeps it from coming apart. If the key is difficult to turn, the lock will need attention as once it binds up and won't turn, it is a real mess to cut apart.

Have you removed the circuit board from the housing, installed a battery and checked for output? Perhaps the housing is holding the switch on.

If the remote has gotten wet, ( even in a pocket on a sweaty day ) minerals / salts can build up on the circuit board causing all sorts of unintended operation.

These remotes have a battery check function that flashes red visible light after the button has been held for 5 or so seconds. Do you have visible light when the IR LED is in operation?

Near as I can tell, the older remotes like yours are not locked to the car so getting a used one is possible. ( 97 + has a RFID chip that locked to the anti theft box and ECM, if the chip is transferred a 97 can accept a used remote. ) The sync procedure is 2 clicks of the remote, insert key into ignition then turn the run position.