Hi JCF,
Sorry for delay with answer.
I think the bluetooth module is dead
Typically they dying if something was shorted or it could be killed by static.
Now you need to replace the module.
Cut it with a pliers from the board, next you will need to remove carefully remaining pins with a soldering iron and install 2.54mm pitch socket ( or solder new module directly to the board)
I recommend you to install a socket, becuse you can buy fake module or it culd be damaged again, so the socket makes replacing more easy.
There is a lot fake HC-06 modules on aliexpress and other shops and I not recomend to use it. It will be better to by JDY-33 or JDY-31 module.
Or if you can by genuine HC-06 - you can buy and use it.
Please write about result after replacing the module.