First of all, when you adopt a pet, you accept responsibility for them for their lives. You don't just "get rid" of them for 1 accident.
I have moved several times over the years with cats and have never had a problem. Make sure you take her litter box and place it someplace easy for her to get to. I always stick the cat right in the box as soon as we get in the door, then let the cat explore from there. The last time we moved my 3 cats found hiding places for a few days...we didn't see them much, but they all found their food and the box without an incident. Slowly they all came out of hiding. Like another writer said, clean the carpets first, and if there's a basement, wash that down if you can (the basement is probably where other cats may have left their mark, or in the corners of rooms).
For the kiids, make it an adventure. They get to decorate a whole new room! They're going to take their cues from you - if you're stressed and worried, then they will be too.
In general, when you pack boxes, make sure you only put things from 1 room in any box and mark the box with what room it goes in. Then as you unload the boxes, you put that box in the room it belongs in. That way it saves you having to search for things when you get there. I don't know how much time you have for your move...so if you can, do all the cleaning at the new house before you move in, make sure there's tp and soap in the bathroom, because you and your movers will need it the day you move in. Also, maybe something to drink in the fridge so it's already cold. The last move we made, we had family and friends helping, so we bought pizza, beer, pop, etc. for everyone. Spend the money for a rental truck (my husband has rented from Ryder, UHaul doesn't take very good care of their trucks), you'll need it for the bigger furniture, and it allows you to move more at a time. You can have your cable hooked up the same day you move in (I stayed at the new house and directed where things went, started some unpacking, and finished cleaning while everyone else loaded & moved things from the old house). Have the power turned on before the move, DTE will add a 2nd account before the 1st is turned off without a problem (the issue we found was that the other people had interruptable service & we didn't want that, but didn't realize that the central air was hooked up to it's own meter box....when we couldn't figure out why the air wouldn't work, called the HVAC guys & they suggested an electrician to re-wire the air to the regular box. only took a few minutes for them to fix, but glad we found out before moving day).
Good Luck! and stay positive!
D.