Aughnacloy is in the RC parish of Aghaloo. Sadly they have no baptism records earlier than 1846, and marriages from 1832. So you won’t find any record of your ancestor in church records.
I looked for Duris (and variant) households in Griffiths Valuation (1860) but there were none in the Aughnacloy area (civil parish of Carnteel). None in the 1827 tithes either:
http://cotyroneireland.com/tithe/carnteel.html
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 are running an Ulster DNA project and can offer FTDNA testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).