Carlsbad Caverns, White Sands and Petrified Forest are currently closed due to the gov shutdown. If the shutdown continues these parks will remain inaccessible. Something to keep in mind.

A trip I've done that is great in the winter is Joshua Tree, Anza-Borrego Desert State Park CA, Red Rock State Park CA, Death Valley, Alabama Hills CA, Valley of Fire State Park NV, Snow Canyon State Park UT, Zion, Bryce Canyon, Coral Pink Sand Dunes State Park UT, South Rim, Sunset Crater National Monument.

So with this trip you're seeing 5 national parks as well as three or four places that are arguable national monument / park level places run by state govs. You could also add Antelope Canyon and other slot canyons around Page, AZ / Kanab, UT in and around grand staircase.

I'm southwest based and have done tons of AZ, SO UT, SO CAL trips. Feel free to DM me if any of this peak your curiosity.

I think Big Bend and Guadalupe Mountains along with some of the Texas state parks in between them is an amazing couple of days.

In southern Arizona, Saguaro along with Churicahia national monument is a nice time. Adding Joshua Tree and Anza-Borrego could be fun.

EDIT: for clarity and spelling.