"Scotmum, that book you mentioned isn't Francis J. Montgomery's
"A short history of Second Donegore Presbyterian Church, 1788-1988" is it? "
No, Christopher, it is "Presbyterianism in Buckna 1756-1992 by Margaret E Millar, ISBN 09520909 0 2" which is now out of print. A number of copies are available for loan in some libraries throughout the province and sometimes second hand copies turn up for sale (a very helpful book dealer managed to secure me such a copy). Tamyneybrack (and its various spellings) was one of many small, rural townlands of the Braid valley, which surrounded the church, with the nearest large village being Broughshane and town being Ballymena.
Having looked again at the timescale in question and seeing that he did not appear to be in the area in the earlier part of the period, I would be fairly confident that this is indeed the Andrew in question. As Val has the date/place of death for her Andrew, perhaps she also has the certificate which may give some further clues.