Gary,
Woodrow does not seem a very common name in Co. Down. In the 1901 census there were just 2 households. 1 in Belfast (part of which is in Co Down) and the other in Killyleagh. Both households were Presbyterian (so probably Scottish roots).
Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you won’t find statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. Ideally you need to know the precise denomination and have some idea of where the person was born in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
https://www.nli.ie/en/family-history-introduction.aspx
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
Looking at the Rosdavies site, Woodrow appears in the parishes of Drumgooland, Inch & Down, Killyleagh and Killinchy.
http://freepages.rootsweb.com/~rosdavies/genealogy/SURNAMES/W/WoWy.htm
Do you know what denomination your Woodrows were? I’d be inclined to ask a researcher to go through the Presbyterian records in PRONI for the above parishes to see if there’s any trace of the family. If you can provide the researcher with rough years of birth that will obviously help.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).