A birth in the 1780s and a marriage in the early 1800s are both well before the start of statutory birth registration (1864) and statutory marriage registration (1845) in Ireland. To trace Elizabeth’s family you would need to rely on church records. Not all parishes in Tyrone have records for the 1700s and of those that do few are on-line.
You haven’t said what denomination Elizabeth was, though I can see the Kenny marriage was Presbyterian (Church of Scotland), so she may have been brought up that denomination in Ireland, though another possibility is that it was her first husband’s denomination.
I looked at the 1901 census of Tyrone. There were 44 people named Evans there then. The majority were Church of Ireland (ie Episcopalian) though there were 10 Presbyterians and 7 Catholics.
The majority of surviving church records for Tyrone are in PRONI (the public record office) in Belfast. They are not on-line and a personal visit is required to view them. (It’s closed at present due to Covid). Without knowing where in Tyrone she lived, it would be a matter of checking every parish in the county, perhaps starting first with those where the name was found in 1901. (That’s the oldest complete Irish census).
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net