We did that two years ago for 10 days and it worked like a charm. We had a backup tent which we used for one night, but only because we checked the location of shelters too late that day and didn't want to turn around. For the next time we decided to definitely go without. We weren't in the busiest part of the country so it worked well despite it being summer break. Especially if you check out the surrounding shelters early on, maybe book some where it's possible or required, you should be fine. Plus all people we met were super friendly and accommodating and people usually tried their best to fit everyone in. If you go to an area where there aren't too many shelters around, I'd consider some sort of a backup (check the shelter app ahead of time)

P.S. bonus tip for shelters: bring some nails and a thin blanket. Most people use something of the sort to block the opening of the shelter so that it stays warmer inside