Damia’s maiden name is from the birth record for their son Alexander.

1840 census of Bath Co., Ky, east of Slate Creek, lists on p. 81, line 16:
Josua Sorril - 2 m < 5, 1 m 20-29; 1 f 5-9, 1 f 20-29

1850 census of Bath Co., Ky, division 2, enumerated 1 Aug, lists on p. 21 as family #281:
Goshua Sorrell, 37; Damia, 32; Phebe, 15; Thomas, 14; William, 12; Ruth A., 10; Martha, 7; Elisha, 5; Shilo, 2/12. All born in Kentucky.

1860 census of Bath Co., Ky, Owingsville P.O., enumerated 13 June, lists on p. 41 as family #289:
Joshua Sorrel, 45, showmaker; Diama, 43; Ruth, 19; Martha, 17; Elisha, 15; Shilo, 9; Moses, 8; Margareta, 6; Benj., 4; Alexander, 2. All born in Ky.

1870 census of Pendleton Co., Ky, precinct 8, Butler P.O., enumerated 4 Aug, lists on p. 16 as family #107:
Sorrell, Joshua, 55, farmer; Daimey, 51, wife; Alex, 11, son; Ben, 15, son; Sarah M., 16; Moses, 18; Shil, 20. All born in Ky.
(The children are unconventionally entered in order of increasing age rather than decreasing.)

The next household is their son Thomas.

1880 census of Pendleton Co., Ky, E.D. 166, enumerated 9 June, lists on p. 15 as family #125/126:
Sorrell, Joshua, 67, b. Ky Va Va, farmer; Damey, 63, wife, b. Ky Penn Va; Sarah M., 24, dau, b. Ky Ky Ky.

The next household is headed by their son Elisha (listed as Elisa).