This isn't a case of name confusion (ie, nissan.com being owned by Amnon & Uzi Nissan, who run Nissan Computers). You're not running a domain called NYSFair.com because your name is Nathaniel Fair. All the content on your site was specifically to do with the fair. They were okay with this for a while, but now they want to run their own site. What basis do you have to insist that *they* shouldn't be allowed to run *their own* Web site?
I think you need to either accept their offer or make a reasonable counteroffer, but recognize that you can't keep the domain - if you fight it you will lose, and rightly so IMHO. If this were your personal domain and the name conflict was unintentional, I'd support you all the way. But you really didn't have the right to register a domain clearly intended to be for someone else's business, without their permission, and then run a Web site of your own from it.
The fact that you've switched the content over to having nothing to do with the fair probably doesn't help matters any. At this point you aren't even maintaining any fair-related content on the site. You're insisting that the domain has to sit idle. Perhaps they were vaguely annoyed at your trademark takeover from the get-go, but at least you were keeping schedules updated and so forth, so they were getting some value out of it. Now you want them to get no value at all, but let you keep the name. Why? Why do you even want the name, if you're not running the site any more?
If you do want to make a counteroffer, you might ask for some publicity on or involvement with the web site they do create. But really, you should probably take the $1100 and be done with it, while it's still being offered. It's a more reasonable deal than most of this type - usually they just sue you. The fact that they're willing to make this offer probably results from their appreciation for the work you've done in past years. The best you can do is ask for, say, $2000; maybe they'll give it to you. But if they refuse, really, take the $1100.
Sory if this sounds harsh - that's not my intent, I just think you need to face up to the reality of the situation.