In Michigan you can carry a sidearm while deer hunting with firearm, but I don't think you can legally be in possession of ANY type of firearm while bow hunting during the bow season. It is legal to bow hunt during the firearms season though, and as such I don't know if you could carry both weapons (gun and bow) or not.
I've never considered it to tell you the truth.
For the Turkey/deer season overlap I can see wanting to keep some slugs either in the pocket or in the pipe, depending on ones first choice of quarry.