Fuel will not be a problem anywhere along I-20 or I-10, though you will not want to get too low on fuel as it may be 40-50 miles between gas stations in some places with one fairly long gap of nothingness between San Antonio and Van Horn, TX. Speaking of El Paso, whatever you do, do not spend a night at Stary Skies RV park in Vado, NM just west of El Paso, the place is a literal junk yard (as in vehicles missing their engines sitting around), plus Vado has a large cattle lot, and even though I grew up on a cattle ranch, I found the smell a bit strong there. Grocery selection may be a bit more sparse, but any town big enough to show up on the map will at least have a small grocery store, though you may think you are back in the 1950's if you go in them.
As to hiking, social distance sight seeing, etc. there are few good options along the I-12 / I-10 route, a few I might suggest are, Vicksburg Battlefield in Vicksburg MS, most of it is self guided drive through audio tour, though since this is only small RV friendly, you will want to use your TOAD car. Expect 1 to 3 hours to do the tour depending on how many stops you make, or a month if you stop and look at each of the thousands of stone markers. There is probably room in the visitor center parking lot to unhook the TOAD and do the drive through.
If you want to take break for a couple of days mid way, consider a side trip down to Fort Davis about 75 miles off your route, though if you need to be in contact with the world stay at one of the in town RV parks if you want cell signal, you have a few outdoor social distancing allowed options here, tour the old frontier fort, with some restored buildings, (big parade ground in the middle, so lots of spacing, and few tourist so even if you go in the indoor restored barracks, houses, etc. there is a good chance you will have the building to yourself. Also there are several miles of hiking trails at the fairly new Desert nature center http://www.cdri.org/ located just a couple of miles south of Fort Davis, as well as nearby Davis Mountain State park (no cell service). There are gas stations in Fort Davis, though they are not particularly RV friendly, there is even a Dollar store there, and a small family owned grocery store. There are also larger stores including I think a Wal-Mart in Alpine about 35 miles away. Going west from here I can't suggest much along I-10 though others may know of options in southern AZ, etc. If possible I would suggest trying to make the drive from the Fort Davis area to somewhere west of Las Cruces in 1 day as I have yet to find a good RV park option along that stretch of road.
Also be aware you will run into a mandatory Border Zone inspection station west of El Paso when going west, or just outside Van Horn when going east. Typically in my experience it is pull up to the window where the agent will ask a few basic questions, what is your residency, where are you heading, how many people on board, etc. they may also have drug dogs doing a walk around. The last time I was there I think it was a total of one question, I stopped the agent asked where I was heading, I said trying to get home to Louisiana ahead of the early season arctic blast, and he sent me on my way.