Serial number lists are only approximate, though for a newer watch, it should be pretty close. The number you specify is about 1999. https://www.omegawatches.com/en-us/watch-omega-seamaster-300-james-bond-2531-80-00 You can ask when it was serviced last and maybe you'll get an accurate answer. Omega factory service is appopriate for this watch, which is not necessarily collectible as yet. I personally use Nesbit's in Seattle because the Miami service center has never once returned a watch to me that didn't have to be sent back. As for stolen... I would contact your local police department and ask their advice.
Another thing you might try for more peace of mind is to ask the seller to direct you to two or three references who have purchased from him/her before -- ideally these references may be individuals you know something about within the world of watch buying and collecting.