Around here, a lot of houses turn over in the summer because families are doing just what you're doing - switching schools. I'd consider renting, though, until you learn more about your new community and until you secure a job. I'd also find something in a district with a good public school system in case your kids don't get into the private school or it's not right for them or it's too expensive on just one salary).
Try to go with the flow on some things. I'm a planner too, but you have to trust that you can manage in so many ways being flexible.
I would consider lightening your load and limiting your moving costs where you can. Sell what you don't absolutely need or love. You could rent a furnished place if necessary, you could rent an apartment and actually rent furniture short-term, or you could buy new. Lots of options.
Re travel: I would turn the trip into a vacation. What a great experience for your kids! The weather would be hospitable, and you could include them in the planning. I'm showing my age, but honestly, don't miss the opportunity to teach them map and navigation skills - don't do it all by GPS! I traveled from NY to Texas in 4th grade when my dad was transferred for a 4-month project, and we went in the family car with 2 kids and a dog. We packed lunches, bought stuff on the road, and found dog-friendly hotels (not nearly as easy then as it is now). There are also temporary pet care places now while you sightsee - if your animals are temperamentally suited to that. There are also dog walkers through Care.com or hotel recommendations.
Another option to consider is shipping the car and renting an RV. We have taken multiple vacations this way, and our son's college tour as well. It was the same money as renting 2 hotel rooms (or cramming into a single one) and buying all our meals out. We ate out or got take out some, but we also made breakfasts and lunches in the RV, and packed a small grill for BBQing in campgrounds (where we met terrific people and our son went swimming and so on). They're a bit of a pain to pack (you need dishes, pots, silverware, etc.) BUT you're going to be packing anyway. And once you pack up the first time, you're done - much easier than going into a different motel every night then repacking the next day. Our dog had free rein in the RV but we also had the crate for when we went out. And you can leave the AC on for an animal in the summer, which solves the problem of leaving pets in a hot car. Plenty of campgrounds have pull-in/pull-out campsites so you don't even really need to learn to back up in tight places. We packed enough groceries for 3 days, then pulled right into a supermarket parking lot to pick up more, and kept going! So easy. It gave us the flexibility to change our minds and our route without worrying that we have to cover a certain number of miles in order to make our hotel reservation. If you want more info and tips, message me.
If you don't have AAA, get it. They not only help with the obvious roadside problems, but they will help you map out a trip, including sights to see, hotels (or campgrounds!), discounts, and more.