This is a problem to solve all on your own. I've never heard of a person being unable to move their tongue. It is virtually impossible to not move your tongue while actually doing an accent. Perhaps you have no feeling in your tongue? This situation you find yourself in I think can be potentially dangerous. Sleep apnea is a thing. You wouldn't want to accidentally choke on your tongue in your sleep. Perhaps you have a very tight tongue tie? You know the skin beneath the tongue... That could be an option. I'd say see a doctor. Such a thing is odd. Your psychologist can help you on the mental side of things. But I'm guessing this is more physical than mental.