The relay is likely not the problem.
If, when it's turned on, it's generating RF, and you said the relay is working, and the TX light on the radio is on, then it's likely something in the PTT circuit in the radio.
Unplug the microphone and see if it does the same thing.
Then, remove that accessory plug at the rear of the radio. You will lose RX audio, but does it still transmit?
After that, open the radio and see if there is anything metallic that fell behind those connectors.
Use a multimeter and see if the PTT line is shorted to ground at the microphone connector.
And lastly, get a manual or schematic, and follow the PTT line through the radio. There is likely something shorted between PTT and ground (mostly capacitors and transistors) that has gone bad.
It's a very old radio, but that makes it easier to repair and diagnose.