Quick report: Is it feasible to take dry ice on a camping trip and keep your drinks cool with it? Yes, although it's not cheap or particularly easy. The results are however much better than car-powered cool boxes, in my opinion. You'll need to find a dry ice supplier who is either on your way to the campsite, or who will deliver to you before you set off. I found All About Ice, based in Hemel Hempstead, which was handily on the route I was going to take to get to Swanage. I paid 35 pounds for 10kg of dry ice pellets and a thick polystyrene container for them. The container is a cube about 40cm on each side. Needless to say if you want to pick up dry ice you'll need to get there during normal working hours, which probably means a day's holiday from work for you.
The next problem is transporting it - dry ice is constantly subliming into carbon dioxide gas, which will suffocate you in a closed container, and if you're driving you'll probably kill more people than yourself if that happens, so be really careful with it. This means you drive with all the windows open, and if it rains you get wet.
Once you've actually got the dry ice to a campsite, most of the difficult parts are done. You'll need another cool box, as you can't put food or drink straight into the dry ice container - it's about -86 centigrade in there. Put some heavy gloves on and transfer some dry ice into the bottom of a cool box, then cover that in a bit of cardboard and put your food or drinks on top of that. If bottles come into excessive contact with dry ice they will freeze and could shatter. If you need to cool something quickly, you can put another layer of cardboard on top of your beer and then shovel some dry ice on top of that. Now you can enjoy cold beer and wine well into the second and third day of the weekend and be the envy of the campsite. You'll probably only want to transfer a little bit of dry ice into a cool box at a time. I also found I could leave the cool block things you get for cool boxes in with the dry ice overnight without them exploding, but your milage may vary on that.
Lock the dry ice in your car when you don't need it, to prevent children and idiots playing with it - it will burn you if you touch it. Don't forget to give your car a few minutes airing with the doors open before getting in if the dry ice has been in there overnight. Don't put dry ice or any container containing it in your tent, for obvious reasons. Apart from that, the only problem will be disposing of it if you don't want to take the remainder home. Rather than just dumping it somewhere children or pets could burn themselves on it, you'll want to pour a lot of (ideally warm) water on it to disperse it. This could be quite a lot of water, so be prepared for a few trips to the tap.