If you're not comfortable with the material you're playing, and the phrase "paralysed with concentration" would indicate that you're not, then you're not going to feel confident enough to do anything other than just stand there & get through the gig.
The way you rehearse sounds like it could have a lot to do with it, hiring a rehearsal space so you can set up & play in a more "gig-like" way might be worth doing. You don't have to do it all the time but every now & then, especially just before a gig, might be worth your while.
At the end of the day some people are comfortable moving on stage & some aren't. As others have said there is nothing worse than seeing a band who is obviously trying too hard to look like they're having a good time when it's obvious they're uncomfortable with it.
When I used to gig some nights I'd be dancing around over the stage, other nights I'd spend 80% of the gig with my back to the audience, it depended entirely on how I felt at the time. I was confident enough with the music though to know that if I suddenly decided to start jumping around I could do so & not have to worry about my playing.