I searched the Irish Family History Foundation's database. It contains 11,500,000 surviving rcds for abt 60% of Irish counties.
1. For Samuel Carson B 1824 +/- 4 yrs, I found 1 B rcd in 1823 in Co. Antrim.
2. For David Carson B 1819 +/- 4 yrs, I found 1 B rcd in 1819 in Co. Cork. This is a long way from Co. Antrim.
3. For Ann Trotter B 1828 +/- 4 yrs, I found 0 B rcds. The earliest rcd for this name is dated 1870. There is a B rcd for a "Mary Anne" for 1826 in Co. Cavan. This has a slim chance of being for your ancestor.
4. For Dickson/Dixon Carson B 1831 +/- 4 yrs, I found 0 B rcds. This infers the event wasn't rcd'd, or the rcd didn't survive.
5. For William Carson M 1817 +/- 2 yrs, I found 6 M rcds: 1815-Mayo, 1816-Galway, 1817-Antrim, 1819-Galway, and 2 1819-Mayo. The Antrim rcd has the highest probability of being your ancestor. A view of the M rcd SHOULD show the spouse's name. This fact will validate/invalidate the rcd.
Google the Foundation's name to lead you to this database. Pls post your search results to my genforum msg.