Debbie,
You would search under Tullyhugh. Though townland names are not in such great use today they do still feature in street names and you will see them on some signage. But in any case you are searching records for the 1700s and 1800s so the townland was predominant then. A townland is a unit of land. It’s a system unique to Ireland and dates back to at least the Norman times (1066) or earlier.
There were no street names or house numbers in Ireland in the 1700s and 1800s (save in Dublin and one or two other large cities). Your townland was your address. The postman and, anybody else who mattered, knew where in the townland you lived.
A group of townlands is a parish (Ballymore in this case) and a group of parishes makes a barony (Orior Lower in this case). Church records tend to be compiled by parish, but baronies feature in the Registry of Deeds (from about 1835 inwards) as part of their indexing system. As the number of deeds being registered increased, they broke them down by baronies.
For information there is another Tullyhugh in Co. Sligo, so that’s something to bear in mind if searching databases. And just to make it more interesting, some parishes have a townland within them of the same name. So the parish of Ballymore has a townland named Ballymore in it. You often need the context to ascertain which is meant in any particular case.
Tullyhugh hasn’t been renamed Tandragee. The town of Tandragee (sometimes Tanderagee) has sprung up across 3 townlands: Derryallen, Ballymore and Tullyhugh. Most of the town is in Ballymore with just one end being in Tullyhugh. Most of Tullyhugh remains agricultural land. If looking for it on Google Earth, the Madden Rd runs through it.