servicing for vintage speedmaster

Hello, My father recently gave me his old Speedmaster that he said was bought new in mid 1960's. He wore it everywhere and has a lot of personal history. He sold voice recorders to NASA for the Apollo space program so had a good story of meeting Pete Conrad (prior to his being 3rd man to walk on the moon) sitting at the hotel bar out side NASA Langley after a long day and they got to talking because they had the same watch on. Other than that I don't
know much about it. Maybe a 105.012 or 145.012 from what I've read?? It works well and keeps decent time but not sure when the last service was. I have a few names of vintage Omega service experts but none local to me in the Wash DC area. I would feel more comfortable not having to ship it and wanted to see if anyone in the forum could refer me to an Omega expert that can service
the vintage watches. Also what should and should not be done. I've attached a few pics that show condition.
Thanks for any assistance in

thanks for the comments. I had a bad experience with a local shop that seemed competent so that's why im a little gun shy.
Attached pics of the bracelet. I don' think its original but what I can find out its a #12 with #47 end links.

cant thank you all enough for the responses. much appreciated!

If you're comfortable mailing it, there are also many highly respected folks that are mentioned here on the forum.