Skip Albuquerque and Relax
Welcome aboard the RoadTrip America Forums!
I've done that segment as part of much longer RoadTrips between the East Coast (family) and Tucson (home), and I've never gone by way of Albuquerque. There is a much shorter route if you're willing to leave the security blanket of the Interstates. From Amarillo take I-27 a short way to the south and pick up US-60 west through Clovis NM and switch over to US-70 west through Roswell and Alamagordo to Las Cruces where you'll get back on the Interstates, I-10 west to Tucson. That will lower your driving time to about 14 hours. Now, with stops (food, fuel, mental health breaks) this is still two full days, but they'll be considerably more restful. A good overnight stopping place would be Roswell. This will be your first sizeable town since Clovis with enough lodging choices to find something you'll like at a reasonable price. Plus it has a certain 'mystique'. Your second day's drive will include an Indian Reservation (Mescalero Apache), the home of quarter horse racing in America (Ruidoso NM), and White Sands National Monument.
AZBuck
Matters of Personal Choice and Logistics
It is very hard to recommend one site over another as what one likes depends as much (if not more) on the 'one' as on the 'what'. That's why we take great pains around here to avoid the terms "best" or "must see". There is also the problem of the constraints that your trip will put on you in terms of time available, extra miles you're willing to drive, etc. So with those caveats in mind, here's my take on the sites you asked about. White Sands is an incredibly unique spot for a number of reasons. First of all there is the composition of the sands. Most sand dunes (the beach, Great Sand Dunes, and others) are composed of quartz grains. I don't mean to denigrate them, but quartz is possibly the most abundant mineral on the Earth's surface and its production is the normal result of typical rock erosion. The White Sands, on the other hand, are composed of gypsum. So there is the unique geology of there formation, and the dazzling effect of that particularly bright white mineral. There is simply no other place on Earth like White Sands. At the same time, if you take the route I recommended, you will be literally driving by the entrance to the Monument. You will never get closer. One item of additional interest. You may have to stop even if you don't intend to as US-70 is occasionally closed (for short periods) when there is to be a launch at White Sands Missile Range.
And while I have not been to Gila Cliff Dwellings National Monument, I have never been disappointed in any early Puebloan site that I have visited. Many require significant hikes to get to while Gila Cliff seems to be within a mile of the trail head. There are also hot springs in the area that you can hike to. However, it is a bit off your path, and forces you to add a hundred mile drive on a couple of two lane roads besides the detour to get into the Silver City area. I just don't see where you have the time for it on this trip.
AZBuck