Bruce,
Forgive me if I repeat information I maye have given previously, but Church of England (in Ireland) is the Church of Ireland, and so if your ancestor was Anglican those are the records to focus on here. There are 29 parishes in Co Armagh. See map with names attached:
https://www.ancestryireland.com/civil-parish-maps-for-ulster/civil-parishes-of-county-armagh/
You or a researcher would need to go through them all looking for your ancestor. Bear in mind that not all parishes have records (many were lost in the 1922 fire) and of those that do they are not all on-line. Copies of nearly all the surviving records are held in PRONI though. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
But your basic problem is that with only a possible birth in the 1780s and 1790s, and no parents names, I don’t see how you could be sure of knowing which of the multiple John Williamsons you are likely to find is going to be the right one.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).