What should I do about a huge gap under the drywall and baseboard when installing luxury vinyl tile?
Take the baseboard off and install your floor.  Cut your vinyl tile even with the sheetrock, less 1/4 inch -- this will allow space for you to lift your tile and lock it in place; after you have installed your floor, you can reinstall your baseboard to sit flush to the floor, leaving a finished look. The 1/2 inch thickness of the sheetrock will give you plenty of expansion  room. There is very little expansion  in vinyl. Trust me, I spent 29 years installing flooring.