June should be a not as crowed season. Our trip was in June many moons ago. Some campground will not be completely open, but they will have many loops to open up if the need them. IMHO, I'd stay in the parks themselves. At least two sites would be best, 3 would be good as well, might save a little fuel money.
Being less crowded at that time, you should be good without a tow. Don't sweat having a reservation, we have been there 4 times without having one.
Might want to do 3 days for Tetons, but 2 days could work. We have only stay at Colter Bay Campground, the one without hook-ups, it was a nice place for such a large campground and close to showers. Our daughter & her hubby needed them, not having showers in their A-Liner. I have heard good things about Gros Ventre Campground...get more animals walking by or close by, etc. Nice views.
The Colter Bay Visitor Center and Indian ArtsMuseum by the Colter Bay Campground in the Grand Tetons is really a neat one.
Here's an older post with info from our past trips, etc.
DesertHawk's Recommendations for Yellowstone Area.