At roughly 1,850 miles, your trip is too long to try to do in just three days but should be relatively comfortable to do in four days. That means you're looking for three evenly-spaced overnight stops. Those would fall around El Paso Tx where there are plenty of choices, Eastland/Ranger TX where there are a few choices, and Vicksburg MS which also has lots of options. Again, we're flying a bit blind here as we know nothing about what you're looking for in accommodations, but you should find something to suit your tastes in each of the above listed towns.
One other thing... This could be a mind-numbing drive, mile after mile of Interstate can and does lead to a condition called highway hypnosis. To combat this we recommend that you stop every few hours, get out of the car and take some fresh air.
I'd personally drive a bit farther the first day and stop in Van Horn TX. I'd also get closer to Fort Worth on the second day - probably Weatherford which has a lot more choices. That would allow you to get as far as Jackson MS the 3rd day.
A quick way to find options and deals for lodging is to do a search through the Reservations page.
I'm going to agree with GLC's suggestions for possible overnight stops. Van Horn, Weatherford, and Jackson all have choices for hotels. We don't usually recommend specific hotels, and one reason is experience -- it may have been awhile ago and hotels (as well as restaurants) change owners/managers all the time. They also may change brands, too. You can shop for hotels right here on RTA. If you plan to travel a lot, sometimes it pays to stick to one brand, so you can collect points good for a free night or two somewhere. By "one brand", I'm talking the conglomerate; for instance, Choice Hotels includes Comfort Inn & Suites, Quality Inn, EconoLodge and others; Wyndham Hotels include Days Inn, Super 8 and others; International Hotels includes Holiday Inn Express and others; Marriott has a lot of different brands as well.
If you don’t have hotel reservations, the free iexitapp can be very helpful. We have found it extremely valuable searching for hotels, restaurants and gas stations. Oh, and rest areas. This eliminates the stress of reading each passing exit sign.
You select your state, Highway and direction, and exit. It wlll show you virtually everything at each exit. Rarely, hotels will post special deals as well.
So, let’s figure you want to drive an hour more. Search the exits and see which have hotels that appeal to you, and call them. You’ll quickly find out if they’re booked or out of your price range. I’m amazed at how much you can sometimes save by driving 30 minutes further or closer.
It’s also been great looking for lunch or dinner, especially if you have a favorite in mind.
On the same idea of travelingman's suggestion of an app, there is a book called The Next Exit.. If you are like me and have limited data usage, or you don't like to rely on the hope of having a good signal wherever you are, paper maps and books have it in spades over the electronics.
Not that I don't like my electronics -- I use Google Maps in figuring mileages, and research hotels ahead of time on my laptop -- but when in a car, normally I rely on maps and my book.
The one thing that both that app and the book I suggested have in common? They are both limited to interstate highways. If traveling on a US highway or state highway, you will need to read the billboards and other signs.