I feel like I should go on one, but I am not that handy with tools, etc. And I am afraid I would just be in the way. Is that all they do on mission trips, build stuff, or do they do other stuff that I could help with?

Talk to your pastor and ask about what types of short-term missions trips are available from your area. Some do involve building, while others do not.

I recall visiting one church where a missions trip was planned. They were going to help refurbish an orphanage in a 3rd World country. They needed folks with all sorts of skills-sets. One 84-year-old lady told me she was going. I asked her what she was going to be doing and she gave me a surprised look and said "Oh, me? I’m going along to play with all the babies!" and gave me a huge grin.

Check with your pastor or missions-coordinator and see what’s available. You may well be surprised.