With our 40' motorhome we've stayed IN the parks themselves - better scenery and less driving.
Yellowstone - Mammoth, Madison, Bridge Bay. Fishing Bridge is the full hookup park in the park but we haven't stayed there. Dry camping is just fine with us. We've also stayed just outside of West Yellowstone at Baker's Hold national forest campground. Since we've been to Yellowstone so often we sometimes use Baker's to explore other areas outside the park, too.
Grand Teton - Gros Ventre is our absolute favorite IN the park. We've also stayed a Colter Bay campground IN the park. It is right next to Colter Bay RV if you need hookups. Gros Ventre and Colter Bay campground are both dry camping but have a water fill and dump station. Neither one accepts reservations but each has 300 sites which rarely fill. Pull in by early afternoon and you'll get a spot.
Staying in the town of Jackson is too far of a drive if you plan to do much in the park. Also, you definitely won't have the gorgeous scenery.
You mentioned going THIS spring. It's now spring. :) If you're leaving shortly, be sure to check that the roads will be open. Here's the official Yellowstone web site which gives you road information, campgrounds, and things to do. Each national park has a 'nps' site like this. Check them out.
Summer is the only time they can do road repairs. Click on the road construction link to see what's planned for this summer. There are some closures, detours and delays planned.
http://home.nps.gov/yell/planyourvisit/parkroads.htm