Here's my $0.02 on the subject.
According to the Yellowstone website, Norris has 111 sites, with only two of 50 ft and five of 30 ft. IMO you would be very lucky to have someone leave one of those sites on the day you want to arrive and also be there before someone else gets the site. There could be ten other folks waiting for a site when you arrive and you get skunked.
Also, IMO Madison is better located than Norris for the thermal areas of the Park. Norris is pretty much hot springs with a few geysers and IMO much less interesting than the thermal areas south of Madison on down to the Old Faithful area. Lots of geysers and other thermal features at and around Old Faithful.
We have stayed at Norris on two occasions, including last year, but we are always there in early June when it's less crowded (we have never been there in July).
My suggestion would be to reserve at Madison so there is no worry about getting a place to stay. I also suggest trying for Mammoth for a few days so you can visit the Lamar Valley, which is the most remote road-accessible part of the park. Large herd of Bison there each time we have visited. You can always make a day trip from Madison to the Yellowstone Canyon to see the falls.
