Yes, you definitely need a dress watch. Sounds like perhaps a Nomos Orion could be on your wish list too. Though as used watches don't seem to be an issue for you, you could get something really interesting for less dough with a bit of hunting. My Omega has family history, but on eBay would probably go for about $300. Go up to $1k and thereabouts and all manner of wonders are there.
1K would be way too steep for me and I'm not sure omega would be the right timepiece for me right now. I'm going to be applying to medical school and I'm a bit weary of showing off labels. I'd be worried that someone would notice the watch tag. I want to be dressed properly, but I don't want to look even slightly ostentatious.
I think a vintage omega seamaster would fit the bill though. Isn't that what you have? (trying to figure it out from the picture).
If I can't find anything in my range, I'll probably just get a Seiko or Orient, to be honest. They're great bang for the buck in automatics.