I tried to trace this family through the censuses to see if later ones gave their actual names, rather than just initials.I couldn't find them in 1860 in the same county, but I think this is the family in 1870 and 1880:
1870 Texas, Van Zandt County, Page 144.
Dwelling 1011/Family 1163
Joseph W Price, 50, white, M.D., $300 personal property, b La
Adelie, 26, female, keeping house, b Oh
John, 7, b Texas
Syyntha, 3, female, b Ark
Thomas ?r., 7, b Texas
David, 13, b Texas
Ar?? Short, 50, female, at home, b Ohio
1880 Texas, Van Zandt County, ED 122 (precinct 4), Page 88D.
Dwelling 124/Family 149.
J.W. Price, whtite, male, 62, head of household, farmer, b Ark, parents b VA
Adelia, white, 42, wife, keeps house, she and parents b Ohio
D.A., male, 22, son, b TX
John, 16, son, b TX
Cynthia, 13, daughter, b TX
Thomas, 10, son, b TX
Ophelia, 7, daughter, b TX
Anna, 5, daughter, b TX
Preston, 1, son, b TX
I also checked the Texas Marriages db and found the following marriages that may belong to the children listed in the 1850 census:
Caroline Price married John Watkins on 13 Jan 1861 in Van Zandt County
Synthia C. Price married A.H. Bunch on 4 Dec 1865 in Van Zandt County
Susanah Price married Thomas C. Benton on 3 Jan 1866 in Van Zandt County
Nancy Price married William Bunch on 21 Nov 1866 in Van Zandt County
Based on the ages of the children in the 1850 census, it looked like the family may have been living in Louisiana in 1840.There was only only 1 J.W./Joseph/Josiah/etc in the 1840 LA census who matched the ages:
1840 LA, Claiborne Parish, Page 103
Josiah Price, 1 male <5, 1 male aged 20-30, 1 female <5, 1 female aged 20-30
Good luck finding more information about this family!