From personal experience (a BAD experience) you should plan on putting anything you get in the boat. Trying to paddle a kayak straight while dragging a body behind it will prove more difficult than schlepping it through that mile of spillbanks. If you don't field dress it, it will be that much heavier, the body cavity will eventually fill with water and act like a sea anchor and drag you down. Field dressed the chest cavity will scoop water and make it impossible to make any headway or be able to track straight.
If you think you wanna try it, take an aired up spare tire and tie a rope around it and try paddling while dragging that. Then multiply it by whatever size and weight of deer you might kill.
All is not lost though! If it is legal in your state to quarter or even butcher and debone a deer carcass after a successful hunt, most average deer will fit in a cooler you can strap on your boat.
I have a sit on top kayak and have hunted out of it quite a bit. I had a great turkey spot picked out that I was going to access from the river but ran up against some odd regulations (or maybe just an odd interpretation of the regulations) that prevented any access other than "at official, established entry points". Other than that I have deer hunted and squirrel hunted from it a lot. It is a great way to get to places others can't like islands in the river or public land that is cut off by private no access land.
I knew a guy that planned to deer hunt with his kayak. He went and got a 50lbs sack of feed, sealed it up in a contractor grade trash sack and practiced paddling around with it strapped to his boat. I haven't gone that far... but it's an idea.
Good luck with the new kayak. I have really enjoyed hunting and fishing out of mine.
OkKeith