Are there any TMJ exercises I can do on my own?

Definitely chew some gum before bed. This will exercise the jaw and give it some relief.

Eeps, that makes mine worse!

oops, sorry. it always helps me. massage DEFINITELY helps but my massage therapist does it for me and wow…I clench the sh*t out of mine. Maybe try to find the spot(s) where the top and bottom jawbone come together and push like a pressure point??? Otherwise, ask your dentist. He or she will probably have some tips.
I have a really good one and cute too! ;-)

Here is an article with some suggestions.

To alleviate muscle pain due to tight jaw muscles, perform the following TMJ exercise: Slowly, open your mouth as wide as you can. Then slowly close it. Next, slowly open your mouth to the right as wide as you can, then slowly close it. Lastly, slowly open your mouth to the left as wide as you can, then slowly close it. Repeat these stretching exercises at least 3 times a day. Perform these until you begin to experience some relief.