Hey and welcome to the forum!
I don't have the newest Xbox; I have the regular Xbox 360 (the white one) and KotOR runs great. I don't think you should have any problems running it on any Xbox.
I'm actually doing a playthrough of the first game on Xbox and am having a blast. I would recommend though to play TSL on PC if you can so you can use the Restored Content Mod - the Xbox version is an unfinished game.