Hi,
I searched the Tithe records (these are free) and there are 24 James Kavanagh in Wexford. These are in different town lands, Enniscorthy would have town land names like we have streets.
In the registersonline it just states Enniscorthy, no address.
I was member of rootsireland.ie this would be a place I would look. Join for a month as pay to view is expensive. they have most BDM. Don't get any other records on Pay to view as these are in other sites free.
For John born 1816 one of his sponsors was Edward Sinnott, in the Tithe records he was listed as Castle Ellis east of Enniscorthy town.
As you don't have town land names it's a case of digging, sorry.
Maggsie