Is it possible to transfer my packet from one branch to another after being denied a medical waiver during my initial attempt to enlist?

You'd probably still have to have a medical waiver submitted. And if the Surgeon General denied it once for one branch, they may deny it again for another. No real way of knowing until you try it.

MEPS is designed to find anything medically or mentally wrong with you. It depends on what your condition is and why you need a wavier.

Well, that depends on the waiver and the branch. The Navy is the easiest to get into and accepts more waivers than any branch. Other branches are less accommodating. It really depends on branch, MOS, and what the waiver is required for. In processing for a break in service or going to a new branch requires a whole new full physical as well. There are different requirements in a full physical for each branch so it is necessary to repeat the entire process.