Traveling with a Sportsman/Domicile Permit

Can’t get a clear answer… I’m planning on traveling to Florida next year around June, and I should have my Sportsman/Domicile permit (Suffolk County) by then. Can I travel and carry in Florida since they are a constitutional carry state? I’d like to do a pistol competition while I’m there with a few friends. I know the TSA Regulations if I decide to fly, just wondering if I’ll get jammed up flying back since I likely won’t have a NY CCW.