I think I MAY have found your Andrew J Ainsworth family :::
1910 Justice Precinct 2, Trinity co., Texas:
JASPER (middle name?) Ainsworth age 30 b. Tx., father b. Mississippi, mother b. Mississippi
wife Nannie, age 30 b. Tx. father b. Ireland, mother b. Louisiana.
dau. Zelmer age 4 b. Tx, f. b. Tx. m. b. Tx.
dau. Myrtle age 2 b. Tx. f. b. Tx. m. b. Tx.
Jasper listed as a 'traveling insurance agent.'
Census dated April 27, 1910.
This may be why, we have not been able to find him, with him using his middle name of legal purposes.
I am not related to Ainsworth family, but have helped many to locate their relatives. Hope this helps.