I have come across many spellings for McAviney/McAvinue surname in the Fermanagh area. In Fermanagh there were McAvinues in Galloon, Enniskillen and the Derrylin parishes.
It is important to locate the correct townland where your ancestors resided before emigration. I wonder if Dromanary could be Dromany?? There were McAviney's in Drumany More near Derrylin County Fermanagh.
The problem is the lack of church registers for the dates you are looking for. Also there were Corry/Curry families residing in the same area as McAviney's/McAvinues in Drumany More.