Holy cow! 9 cats? 3 dogs? OMG. Bless you. We travel with 2 feline fur babies and that is challenging enough. Why not just do it the old fashioned way and get a map of the U.S and then you can see all the routes at the same time and choose one? The thing is, for me anyway, is that the journey is the fun part of getting to wherever you are going. For the distance you are going to cover is there any sites you wish to stop at and enjoy the areas you pass through? You stated time is not an issue so explore what may be worth seeing on your way to NH.
I use the RVPARKY app when on the road to find parks on our route. We never make reservations but we do call ahead to check availability when we find a place we want to stay. We have never been turned away yet.
As for a trip planner I use the Good Sam one. You need to be a member to log in and use it. The other one that's free is Passport America's' ....camp4less.com. You might consider joining PA as it offers 1/2 price fees at participating campgrounds, great for an overnight stop.
