I always use I-20 to bypass Dallas-Fort Worth. It skims the southern edge, goes through some of the suburbs. At busy times of day it will get congested around the intersections with the North-South expressways and the loops, more so around Fort Worth, which has more sprawl to the south (Dallas sprawls north).
I am not aware of construction right now on I-20, not up on plans for this fall. I-20 should be passable (unless closed by a wreck, which can happen anywhere, anytime). Texas doesn't usually close trunk highways for construction, which tends to take place year-round. They do sometimes close urban expressways overnight. There is not a summer road construction season, it the way there is further north.
Memphis to El Paso, I probably would not get that close to Metro DFW, taking either Tx-34 down from Greenville to connect with I-35 East at Italy, or be on a diagonal route from east Texas to connect with I-35 at San Marcos, to avoid Austin. Congestion for Austin can make Metro Dallas look like a breeze.
US trunk highways and state highways in rural Texas are usually pretty good roads, although you have to slow down for the towns and work out the bypasses for some of the small cities. Another route I've used, TX-19 is a good connector from I-20 at Sulphur Springs to Athens, where you can pick up TX-31 to Waco, mostly divided highway. But from Waco you still get to drive through Austin.
San Antonio, if not a busy time of day, the FM-1604 loop around north and west sides can work pretty well, I used it often when my daughter lived just inside that loop. Other 1604 sections, west and south sides, are slow. I-410 loop tends to be more congested around the interchanges.