I just finished up a three week trip and did all three parks that you mentioned. Here is what I did. Started in Tetons, Yellowstone, Glacier, Battle of Little Big Horn, Custer SP in South Dakota, and then back home. I had a 33 foot trailer so I was limited on campsites.
Tetons: Gros Ventre. Stayed there 3 days. It was first come first serve and on June 4th, the place was very open with lots of sites to choose from.
Yellowstone: I had reservations at Fishing Bridge made 7 months or so in advance. Stayed there for 4 days and would not stay there again. Roads were terrible and either extremely dusty or muddy. If you stayed outside your camper for more than 5 minutes you looked like a powdered doughnut from all of the dust from the high traffic moving in and out of the campground. When I left there, my truck and camper were absolutely filthy. They did have one paved loop so it may have been better. If done differently, I would have stayed in Mammoth Campground. We love the north side of the park more. More wildlife viewing and hiking trails. That is strictly my opinion.
Glacier: After an overnight stay in Helena, we hit Glacier right at 11am for the first come first serve at Many Glacier Campground. That was a bust, it was nearly full and the several spots that I thought my camper could fit in, it would not have worked. So we ended up staying at St. Mary KOA on the East side. They were not full and had plenty of spots. Apgar would have been a close second, but we prefer the hiking much more on the East side of the park than the West side. Going to the Sun Road was still closed, so this would have been a pain in the butt driving around the park to hike the East side each day. In July, is should be open so that will cut down on driving time traversing the park.
So to answer your questions, July will be a lot busier in each park than June. June was busy but was not terrible at all. All of the campground host in each park stated that things really picked up in July/August. Gros Ventre in the Tetons being the exception. Campground host said they hardly ever fill, but the spots for the 30 plus foot campers will usually go first. When I was looking at Yellowstone, for the month of June, Grizzly in the West plus a few others still had openings when I was looking 7-8 months out. For July, you may want to start a little earlier. However, if you don't mind dry camping, hands down I would stay in the park to cut down on driving time. Hope this helps
2019 Rockwood Signature Ultra Lite
2016 Ram CCSB 2500 CTD
GenY Weight Distribution Hitch