Sedo, Moniker and few other places will help you and broker (assuming it's worthwhile).
There are a handful of experienced domainers who broker as well, also domain lawyers do this stuff.
The biggest issue is what ballpark you're operating in. If your budget is <1000, it's not worth bothering with anyone else. Between 1-5k, maybe some individuals will help out/lawyer. As that number gets higher, more people would be willing. Other big factor is expectations, having a 100,000$ budget for sex.com is all but a waste of time too. So the domain and budget need to be approximately in tune.
If you want some specific help on the domain you're trying to acquire I may be able to help you out some (free advice), contact info is in my profile.
Failing that, go to the website (assuming there's one on it), and see if there's contact details on it somewhere.
If that still doesn't work, Sedo are fairly good at ferreting out information.
I'd always try to make the actual buyout overtures myself, as you get a better feel for the person that way, and you never know when the connection might come in useful in the future.
I'll try and get a mate of mine who's a domainer to respond to this with his thoughts.
As an experiment I'd like you to open a nix console, type in whois* domain name and see who owns the domain and who the domain is registered with. Is it with GoDaddy? If it is GD are using their superior market knowledge against you.
Wishlist for startup ~ one that allows searches w/o selling info & negotiates for domains & makes the transfer stress free & transparent.
Then wait and see
tl;dr: the dollar figure is all that matters, your story makes me vomit in my mouth a little as an excuse to lowball.