I live and hunt in Texas and although I haven't been on any "cull" hunts, I am sure you will find some good ones. Do alot of comparison before settling on anything. Now, as for meat care, When you check into your hunt, I would certainly ask the land owner/guide, if they have cold storage facilities.
Field dressing should be done quickly after the animal has been shot, and then transported to cold storage without much delay. I have seen animals field dressed quickly after the shot, and then driven around in the bed of the pickup truck for another hour or two. In hot weather, that is not a good idea, if you plan to eat what you shot. Good luck.