I have the exact same Tudor 7966 that you have right here. My dad sent it to a local watch repair guy that used to work at the Rolex SC.
I suggest if you don't want any hassle and want a smooth process, just go to the official service center, but they might change a lot of the old desirable parts like the crystal and dial etc. just because its worn and they will not return the old replaced parts back to you.
The other option is to send it to a local independent watch repair shop. Many vintage watch collectors opt for this because they prefer to keep their watches as original as possible and keep the old original parts if it really needs to be replaced.
ultimately, its up to you.