Like Emily, my kids are at public school, but (also like her it seems) we chose the area to live in based on its excellent schools. We got a quick proof of residency by getting a letter from the utilities company confirming we had signed up at that address - so we didn't need to wait for the first bill to prove residency. Our schools have a big emphasis both on academics and sport, and what's interesting is that in the high school, you have to maintain a certain minimum GPA to be allowed to remain on sports teams, no matter how much of a star player you might be. I think that's fairly common across the US.
It helps if you get school reports, but also bring something showing equivalents of US grades to your home country grades if you do. Coming from a different calendar school year there will be adjustments. (We moved when our kids had just finished their school year, but the US was halfway through its school year.) A good school will have good counsellors who take the time to look at your kids individually and make recommendations on that. The younger the children are, the more likely this is to be based on age, but not entirely so. We ended up with one kid being put a half year ahead of where she had been and one half a year back (this one is the youngest kid by far in her grade, though.) As the kids get older and choose their subjects, you might find as we did that moving during the year means you don't get all your first choices, though this was only for one semester so it didn't matter in the big scheme of things.
If you move during the US school year, you do indeed run the risk that the closest school is full and that your child might be put in a different school (depending on the system, possibly permanently or possibly moving to the "right" school the following year). This is why IMO it's important to choose somewhere where all the schools in the system are good, not just some of them. And if your kids are in elementary, make sure you check out the middle and high schools too... Greatschools.org is a good resource. There is massive competition in the real estate market where we are over summer as people try to get into the school district, so this might be something to keep in mind too! We didn't have as much competition moving mid school year, though inventory was less too. Again this is not necessarily something you'll see everywhere, but what is a generality is that good school areas will have a premium in property prices.
Also, make sure you bring the vaccination worksheet for the kids that the panel doctor gave you in the medicals. This is in a format the schools understand re checking that the requirements are met.
Hope that's useful.
it might not be you or your experience, it might be a shortage of opportunities.