Rosie,
Donaghadee is in the parish of Donaghadee.
The 1862 marriage certificate tells you that David’s father was also named David and that he was a farmer.
A birth c 1833 is long before the start of statutory birth registration in Ireland (1864) so you won’t find a birth certificate for him. You might find his baptism but to do that you would need to know his precise denomination. You mention Church of Ireland. That’s possible, so also is Presbyterian. A large percentage of folk around Donaghadee were Presbyterian (reflecting their Scottish origins). Bridget McDonnell strikes me as a Roman Catholic name. It’s possible therefore that this was a mixed marriage and they married in the Church of Ireland for reasons of convenience. Neither may actually have normally been that denomination, though they might have been.
The Church of Ireland records for Donaghadee start in 1771. There was a second church in the parish at Carrowdore but unfortunately its records before the 1870s were lost in the 1922 fire in Dublin. Donaghadee also had a Methodist church with baptism records starting in 1815. Donaghadee 1st Presbyterian has records from 1793 & Millisle Presbyterian start in 1773. There’s a copy of all these records in PRONI in Belfast (the public record office). I don’t think they are on-line anywhere so far as I am aware. A personal visit is required to view them. Researchers in the PRONI area: http://sgni.net
There’s a few Crawfords from Donaghadee on this site here:
http://freepages.rootsweb.com/~rosdavies/genealogy/SURNAMES/C/Crawford.htm
If David Crawford senior was a farmer and alive in 1863 you would expect him to be shown in Griffiths Valuation. There’s no David listed, so perhaps he was already dead. There were 5 Crawford households listed in the parish: in Herdstown (not a farm); Killaghy (a 15 acre farm) and a labourers cottage; Ballyhay (2 labourer’s cottages plus 5 acres of land). So your David might be connected with either Killaghy or Ballyhay.
I had a look at Crawford in and around Ballyhay in the 1901 census. The majority were Presbyterian, several were Church of Ireland and there was 1 “separated Christian.”
http://www.census.nationalarchives.ie/search/
http://www.askaboutireland.ie/griffith-valuation/index.xml
You mention Fullerton in the title of your post but give other information. All I can tell you is that it was a common surname in the Donaghadee area. That vast majority were Presbyterian.