Omega Seamaster KM6736

Hi,
New to the Omega forum, but long time Omega fan. My wife gave me a watch that was her grandfather's. It's an old/vintage Seamaster with what I think is a model number inside the back of the case (KM6736). Definitely solid 14k gold case and band. Movement is an Omega 563 (Automatic; 17 jewels). It literally doesn't look like it's ever been worn, or at least not very much at all. I'm trying to figure out a) exactly what it is; and b) what it's worth. Anyone out there familiar with this vintage Omega?

Thanks so much for replying. Very helpful and interesting. I checked the band and it's 14k as well. The clasp has an Omega on it, so it must have been original to the watch.

Unfortunately, the watch does not fit me. As for service, it's really not needed. The inside is so clean and pristine and the watch appears to work very normally. I took it to a local jeweler/watch dealer who happens to specialize in Omegas. They said that unfortunately it's not of a style/type that is marketable. They offered me melt value (+/- $4,000). They have a pre-owned, excellent condition "No Time To Die" ("NTTD" - which is a watch I've been wanting) which I might use mine to trade them for.