The reason you can't find much info on that is that it would be highly sub-optimal to try to enjoy both Teton and YS from one campground - too much driving. Stay in each. Fish Bridge is the option to beat there. If not there - have a good reason. It's central and convenient.
Find a place to stay at Teton. Where do you plan to enter? You may be able to drive in through Teton - get the "lay of the land" then go into YS and stay - then go back to Teton at the end with a better understanding of what you want to do.
Unless you have a condition requiring hookups - I'd consider a 2-3 night stop at Teton at the end and dump at the end.
I'd highly recommend a good guide for each park and start with 5 nights for YS and 2 for Teton then tune from there based on your interests.