I don’t know what your rights might be without reviewing your donation agreement and the naming rights provisions.  It would be likely that the agreement would terminate the naming right opportunity if the donations stop.

The use of your picture without your permission is probably a different question.  The agreement may be silent on the use of your picture, but you probably have at least a common law right to copyright protection for the artwork that you created.  In the unlikely event that you registered your picture for federal copyright protection within 90 days of when it was taken or otherwise before the infringement began, you could sue for “statutory damages” up to $30,000, plus attorneys’ fees.  In the more likely event that you didn’t register the picture for copyright protection, you could still sue, but you would be limited under federal copyright law to actual damages suffered from the shelter’s use of your copyrighted picture.  As much as you may love your cat, it isn’t likely that you could sell or license the cat’s picture for a lot of money and I wouldn’t look for a grand recovery here.

You might nevertheless be able to force the shelter to stop using the photo and give it back to you.  If you can’t work out a return of the picture on your own, find a lawyer familiar with copyright law to see whether you have any rights under federal or state law.  A cease and desist letter might get the picture taken down if the shelter doesn’t want to be bothered with a controversy.  It is easy for them to get another picture and for most of their audience, a different cat could be just as cute as yours.