Hello Janis,
There's no sign of David in Sligo but there are two listed in the Valuations for Co. Mayo. The Valuations are on John Hayes failteromhat.com website.
Griffith's Valuation extracts - Co. Sligo - 1858 - Culkin
Richard, Bunree, Kilmoremoy
Thomas, Chapel Street, Kilmoremoy
Henry, Carrowmoran, Templeboy
Richard, Abbeyhalfquartee, Kilmoremoy
Margaret, Abbey Street, Kilmoremoy
Richard, Carraun, Castleconor
Richard, Newtown, Castleconor
Patrick, Dooyeaghny or Cloonloughan, Castleconor
Griffith's Valuation extracts - Co. Mayo - 1856-7 - David Culkin
David, Carrowleagh, Kilgarvan
David, Graffy, Kilgarvan
There are three items you require when searching for Irish ancestors.
1. Denomination.
2. Parish
3. Townland.
You'll have to rely on Church records as civil registration in Ireland was not introduced until 1864. Non Catholic marriages were registered from 1845 onwards. When registration for BDMS started in 1864 some people didn't bother with the registration for a few years after that date. Keep your fingers crossed that church records still exist.
Christopher