I remember how overwhelming the paperwork seemed when we were moving here. Mostly it was MUCH easier than it seemed like it was going to be. But do take suggested deadlines and polite requests from people at the university seriously. There can be a lot of back-and-forth through the mail. If your kids don't yet have passports, better get those right away. I was terrible about getting things done on time (3 little kids! moving! selling the house! storage!), so it was down to the wire for us, my husband almost couldn't start his job on time. It took longer for me to get the kids' passports than I thought it would, and I hadn't realized that my husband's company couldn't submit his permit application until it had all the family documents, including passports, too (they asked me a couple times for the documents, but so politely that I didn't understand it was urgent). If you haven't yet, you might start by taking the kids to a passport-photo place and getting lots of passport and visa-sized photos of them (and you, and husband). I assume you'll need visas with photos? The requirements are really picky about the exact size and background color, and I think the size for passports might be different from the size for visas, and you need more of them than you think--to get passports and visas in the U.S. and then upon arrival here, I think, when you register in the community. I think I ended up needing about 10 photos of each person.
Also now is a good time to compile a binder with all the official documents you can think of--"originals" and copies. It will make your life easier in the long run. Try to order two or three "originals" of things like birth certificates and marriage certificates (as you may or may not be aware, weirdly, you can order multiple "originals"). Also make maybe 5-10 good color copies of everything for what you're doing now, and to bring with you--I seemed to be constantly having to run back to Kinkos to make 2 more copies for some other purpose. Here in Fribourg, I once had to bring a copy of our marriage certificate to the post office in order to pick up a package addressed to my husband, to prove we were married (we don't have the same last name).
We lived in Illinois before we moved here. It might be different for you because you're coming on a different kind of permit that we had, but for us we had to go down to the embassy in Chicago in person apply for, I don't know, visas to be stamped into our passports maybe? But I remember thinking that we were so lucky to live close to a Swiss embassy since we had to go in person to the embassy (and not just to drop in, had to have an appointment). There aren't very many Swiss embassy locations in the U.S., each serves a large region, and people in Switzerland don't necessarily realize how hard it could be for you to get to the applicable embassy and may not factor in enough time--places in Switzerland are much closer together. If the embassy is a couple states away, given limited hours, you might have to make an overnight trip (I can't imagine that would ever happen here in Switzerland), which would surely be difficult to schedule. So you might check now if you'll need to go to one (and, if so, if you'll need an appointment, and if even the kids have to go in person) or if you can do your process through the mail.
Even if you don't have to go into an embassy, you might start calling the Swiss embassy with your questions. They'll probably be even more helpful than people at the university because it is what the embassy people do all the time. I found them to be really nice and helpful. Really, if I were you, I'd call them right now rather than keep trying to make sense of the requirements yourself. The embassy hours tend to be limited (probably closed at lunch, open late, close early, and they are probably closed on days that will seem utterly random to you because it is some Swiss holiday you never heard of before), but if you can make a good contact there, you'll get definite answers to all your questions and it will be much faster and easier than figuring it all out on the internet.
Also, you may want to start gathering medical records for your kids. If you've been in the same area since they were born, you might be used to relying on their pediatrician to keep track of vaccinations and to have in his/her computer all the information about important medical events. Parents seem to be expected to do more of that themselves here, and doctor's files seem to be on paper more often than on the computer. You'll want good files to give to your pediatrician here--as you would if you were changing pediatricians within America. (If you'll be living in the town/city of Fribourg and want a recommendation, let me know, as there is at least one good pediatrician fluent in English.) Some of the vaccinations are different here, so you might want to make sure your family is completely up-to-date in the U.S. before you come.
Good luck with everything! Just do one thing at a time. But try to get something done each day.