For the polished area of your watch you can do it yourself easily enough. Maybe you know already but I’ll throw it out there anyway.
I was looking for something for my Submariner for some fines scratches and while reading the Rolex Forum I saw a lot of folks use and recommend Cape Cod polishing cloth. I bought a can and I have to say it works marvelously on fine scratches. It’s a very mild abrasive perfect for such work and doesn’t remove much material at all. You can’t ‘dish’ the steel with this stuff it’s so mild.
I give my Sub and a couple other pieces a rub down every couple years to refresh them. A can will last for years. One caveat, don’t use it on textured steel only the polished.
https://www.capecodpolish.com/