There may be 2 reasons why you can’t find the Moore/Carr marriage in 1820-30. One is that the records are not on-line. The other is that they don’t exist, having either been lost or, occasionally, never kept in the first place.
If Alexander was born in Stranorlar and was either Presbyterian or Church of Ireland then the good news is that there are baptism records for that period. The Presbyterian baptism records start in 1821 (marriages only from 1846). The Church of Ireland baptism records start in 1802 (marriages 1821). There’s copies of both sets of records in PRONI (the public record office) in Belfast. A personal visit is required to view them. They are free to view.
Farmers tended to stay put. There’s not a lot of sense in putting your energies into improving land and then moving every few years. So it’s odd that one child was born in Stranorlar and the other some distance away in Tyrone. However perhaps the mother was visiting at the time of the birth.
Farmers should normally appear in both the tithes and Griffiths (assuming they were alive of course). The tithes for Tyrone are not on the NLI site. They are on this link but you have to search parish by parish.
Or you can search by surname in PRONI (if you go in person). They have them sorted alphabetically there.