The road south through Yellowstone into Grand Teton Nat'l Park is just fine. However, if you're planning to stop at Old Faithful on the way with your 37' you might want to get a very early start in the morning to get in the parking lot before the big crowd. It would also be a wonderful time to experience Old Faithful without the crowds. Be sure to get a 'projected' schedule from a Visitor Center of when the geyser explodes. It won't be an exact time but it will be close.
Then continue south into the Tetons. The only full hookup parks are Flagg Ranch (I think it goes by another name now) and Colter Bay RV. Of the two, Colter Bay is the best. If you can do without full hookups, right next door to Colter Bay RV is Colter Bay campground. It is dry camping. Then midway through the Tetons is Gros Ventre campground. It's closer to the town of Jackson. Both of the dry campgrounds have 300 non-reservable sites. We've always gotten a site with our 40' motorhome.
You'll enjoy the Tetons. It's altogether different from Yellowstone - like a miniature Alps. There are wonderful short easy hikes and a gentle float trip down is Snake River will almost guarantee you seeing a moose in the water with you!!
Just south of the town of Jackson are two large grocery stores to restock your pantry. Albertsons is on the left and a little beyond is Smith's. Within Smith's complex is also a fuel station, liquor store and laundry. What more could you want? Have a great trip!