South of I-10, east of I-35, is all OK in January provided you have power to keep the furnace running. I am comfortable anywhere in this area, and to some extent also the East Texas forests further north. In these parts of Texas you can expect temperatures above freezing most of the time, short freezing spells that won't hurt a eated 3 season RV.
The further south, more likely it is to be warmer, coastal areas less likely to freeze.
Hill Country south of i-20 can be riskier, with freezing temperatures more often and longer, though above freezing most of the time.
All these areas, I will take, havebtaken, my RV in winter. But camping, would't try that without checking for a warm forecast for the time I'd be living outdoors. Used to winter camp, sleeping on frozen ground in Canada, but too old for that now. Don't pitch the tent unless going no colder than 50 at night.
I find San Antonio a fabulous place to visit, usually go spring or fall. I like Houston, Galveston, Rockport, Mustang Island in winter, though more recently I've been going to the Mississippi Gulf Coast or west central Florida at that time of year.
Further west in the high desert (including the Big Bend) and in the southern Rockie, more likely to experience extended freeze. North Central Texas and the Panhandle are subject to invasion by cold continental air masses, think Dakotas winter conditions, just for short periods December-February. But it can get very cold (sub-zero) for extended times, several days to a few weeks.
Texas is a big place, has room for different climates:)
Safety? I know some neighborhoods in Houston I wouldn't wander at night, there are probably the like in the other large cities. In border cities you might not be wanting to try to cheat the drug dealers. From Chicago, you likely understand the big city safety issues, people who get in trouble are usually looking for it.