Four weeks is a good time frame for this trip. You know your kids. My kids have survived three such trips and they were 4 and 7 for the first trip and 12 and 15 for the last. All national parks and similar for sight seeing.
I would heed the warning about Arches. Go through Sedona AZ instead. Add Four Corners and maybe Zion. I would add the Hoover Dam. That is still on our list. I agree with Paw Paw and Gram about Carlsbad Caverns. I don't think the detour north is too bad. You end up in Northern AZ anyway.
Where are you in FL? That will affect whether it's worth talking time on Gulf States on the way over.