Alison,
The date of Joseph & Elizabeth’s marriage looks to me to be 29th Jan 1852, rather than 28th.
They married in Bushmills Presbyterian church. Tradition was to marry in the bride’s church, so that’s possibly where she was baptised So that may give you Elizabeth’s mothers name (and you may get her siblings from that too). That church’s records start in 1820. They are not on-line anywhere so far as I know, but there is a copy in PRONI (the public record office) in Belfast.
However there were no Gillespies in Drumnagee in Rev Traill’s 1803 census of the townland nor in the 1803 agricultural census, so it looks as though they moved there after 1803. Daniel is listed there in 1824 in the tithes so that narrows it a bit.
http://www.irishgenealogyhub.com/antrim/tithe-applotments/ballintoy-parish.php
The Tullybrick that you refer to. Is that the one in Co Derry near Magherafelt? (That would be about 40 miles from Drumnagee). If that’s the location, then the parish is Ballynascreen, and if you think the family was also Presbyterian then Draperstown Presbyterian only has records from 1837 onwards which is probably too late for you. The local RC church has records from 1834.
There were 8 Gillespie families in Tullybrick in the 1831 census (headed respectively by Mary, Daniel, Henry, 2 separate Johns, Thomas, Bernard & Francis). All were Catholic. Here’s Mary:
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Loughinshollin/Ballynascreen/Tullybrick/1/
The geographic distance and different religious denominations would make me wonder if the 2 families are related but it seems your DNA results suggest they are. However you may struggle to establish the precise relationship due to the general lack of records at that time.