Does getting braces mean you have to change your sleeping position?

Braces and sleep. In general no. Sometimes face/mough/jaw can be sore at times which may be a factor in position you sleep.
No. Braces will not change your favorite sleeping position. Some orthodontists still use headgear, but personally i find I do not have to use them, ever, and therefore it is not a problem.
Braces. Braces themselves usually doesn't require you to change posture when sleeping unless you have headgear.