Can I have both barrels of my double-barreled muzzleloader loaded during a legal season for muzzleloading firearms?

Yes. A hunter lawfully hunting small game with a double-barreled muzzleloading shotgun may keep both barrels loaded.
A hunter lawfully hunting big game with a double-barreled muzzleloading firearm during a muzzleloader season may load both barrels with powder charges and projectiles, BUT may have only one barrel or breech capped or primed at a time while hunting.
A muzzleloading firearm is considered loaded if a powder charge and a projectile (either shot or single projectile) are in the barrel AND the barrel or breech is capped or primed.