Its all really very easy to fly with your guns/bows and bring back meat. Guns are super easy. I'm sure there are plenty of threads about flying with guns, so I'll address the meat portion only. It will only be stressful if you let it, and don't do your homework. You will pay extra fees for more bags or overweight bags, but you're paying for convenience.
I've flown with meat many times (deer, caribou, hogs, fish, etc.) . Here is my suggestion:
Before you leave check with your airline and see what their regulations are for excess and oversized baggage. Make sure you know what they are going to require or expect from you. Plan to comply in every way. Plan to combine baggage with your buddies to reduce the fees.
Next once you are there, purchase some coolers from Walmart or other store you can find... The med/lrg $40 chest types are usually pretty cheap and work well. Once you kill and butcher, get the meat frozen if at all possible. Fill up the coolers with the frozen meat (well chilled at a minimum if its a short travel time). Make sure all the meat is sealed in a heavy duty garbage bag (or two) inside the cooler. Make sure you don't exceed the weight limits. DO NOT ADD ICE (Sealed freezer packs are fine). Close the coolers and wrap with duct tape (lots) all the way around it, so it cannot come open with pressure changes in the air. Label it very well with your name, phone number and "date packed".
When you get to the airport to check in, tell them you need some "perishable" labels. They all have them, but might have to look for them since its not an everyday item. If they look at you stupid, as for someone else who is more experienced. Explain you have meat (I don't mention wild game), and that its frozen, sealed, and sealed again. They really prefer you don't use dry ice, since it makes more labeling issues for them and since you don't have vented coolers. Once they add the checked baggage labels and the perishable labels, fly home as you would on any other flight. Pick them up at the baggage claim. It will still be 98% frozen,and ready for processing.
Easy as heck. The time it takes to drive isn't worth it at all IMO. Flying is so much easier that the headache of trying to drive.
Here's an example for American Airlines... You can check up to 10 total bags per person. 50lb limit per bag unless you fly first class (then 70lbs). Coolers are considered regular baggage and not "specialty or sports"). They will accept up to 100 lbs with a big $200 fee. Its cheaper to do multiple 49.9lb coolers.