There aren't many out there, and most adults I have heard of try to compete AAU if they want to go to more than one meet a year. However, a lot of states don't really have a lot of AAU comps. The biggest meets I've heard of are the masters meets. Here are the only ones I know of:
Empire State Games in New York
one in Japan (not sure of the name, but someone at my gym goes to it)
Masters Meet at Azarian Gymnastics in CA
Midnight Madness Masters Meet hosted by Vancouver Phoenix Gymnastics in Canada
Unfortunately there really aren't many opportunities for adults to compete, so it would definitely take a fair amount of advertising to get the word out. On the plus side, I think adults are probably more willing to travel farther distances to compete.
I'm relatively new to the gymnastics world (been in an adult class for about a year), but I would be willing to help get the word out to adult gymnasts if it meant I could have another chance to compete :]
I don't know if I can help you out very much as I'm not particularly knowledgeable myself, but the only Adult gymnastics meets I've come across are the Masters meets. I live in Ontario, Canada and only compete at Level 5 Invitational Level so I'm not particularly high up. I am 18 so I could compete in Masters if I wanted to, but some meets don't offer them or if they do, are at the very end or beginning of the weekend and inconvenient for myself and the coaches who would only have to travel for one day otherwise. So I mostly compete in the 15+ categories.
In terms of competition, the few I've seen just put all the 18+ people in one flight and give them medals based on their overall scores. There aren't enough people to do it by levels so only their scores (not their levels) are compared. I watched a Level 7 compete 'against' a level 3.