I am currently shooting a 7" brace height with a 27" draw. My question is, if I change to a 6" brace height bow, will my draw length change to 28" since I will be drawing it back an inch farther or does it always stay the same?

Your draw length will stay the same (because your body doesn't change). Your power stroke will increase by an inch.

What BTM said. The arrow just stays on the string for an inch longer, thus being a little faster. The downfall is that the longer the arrow is on the string, the greater the chance of bad form or bow torque playing a factor.