If she doesn't want to turn over her interest to you, then she should be responsible for her half of the maint. fees. You should notify her of this by certified letter, and if she refuses to pay her half, sue her in small claims court to get them. Maybe then, she'd be willing to sign it over to you, or accept it from you, or sign to allow it be sold on the open market. If the situation drags on, you can take her to small claims court each year she doesn't pay her half of the fees, if necessary.
You could retain an attorney and have a judge order her to either allow it to be sold, splitting any proceeds of the sale (50 cents?), sign over her interest to you or make her accept your interest in it.
On the other hand, if you used the timeshare, then by all rights, you should pay the maint fee for that year. (Or vice versa.) And if you rent it out, and expect her to pay her half of any fees, then you should be splitting any proceeds you might get above the maint fees with her. So, if you rent it out, to avoid that possibility, you might want to simply rent it out for the exact amount of that years maint fees, so there are no additional funds in limbo that she might counter sue for. Or you could just be a nice guy, rent it out at market prices, and send her her half of the proceeds after deducting that years maint fees. Keep good records no matter what you do.
What an ugly situation. It seems like not getting married to her was probably a good idea, no matter who initiated the break-up. My completely unfounded guess is that she has friends or family advising her that she shouldn't sign anything, which is incomplete advice. She shouldn't sign anything without understanding what it is and having an attorney review it.
-David