It may well be that James Walter Abernathy was Lizzie Irene Evans' 2nd husband, and for her children you need to look elsewhere.
From the ancestry.com's online database of Texas Birth Records (1903-1997) it seems that Robert Archie Dawson and Lizzie Irene Evans had at least two children, both born Smith County TX: Randa Fay, and Robert Lee Dawson. And according to the 1930 census, James W. Abernathy & wife Minnie Evans had 5 children. possibly more in the next decade leading up to the War.
I don't know if there were deaths due to World War II, but it's certainly likely and I would guess that James Walter Abernathy and Lizzie Irene Evans didn't marry after the war, so they may not have had any children of their own. None can be found for the couple on the Texas Birth Records, at all events.