John,

I don't know about shotgun only seasons, but several states are shotgun or muzzleloader only for big game. Illinois is shotgun only. Wisconsin is shotgun only except in Shawano. Illinois is either shotgun or pistol cartridge carbine only, but it has recently prompted the development of Indiana only rounds for deer hunting in the state.

That said, fully rifled shotguns are 100% legal in all shotgun only for deer states, and they are referred to as rifled-barrel shotguns, and not as double rifles like in the UK and some other countries. This particular gun would be 100% legal with slugs for deer and other large game in any shotgun only state, while double rifles designed for what are traditionally regarded as rifle cartridges, even BP cartridges, would not.

Good hunting!