Colin,
You say that William & Esther married c 1821. However their eldest son James was apparently 19 in 1834/35 when they emigrated. That suggests to me they may have married around 1814/15. But who knows?
There is a record of a John Dick marriage to Janet Ramsey in Wemyss on 19.1.1789. That’s viewable on the Scotlandspeople site. Wemyss is a parish on the south side of Fife. Latterly a coal mining area. There is also a record of 3 children born to that couple:
David Dick 1.3.1794 (probably died young as name re-used)
David 5.4.1795
John 5.3.1797
All 3 births were in Auchtermuchty. Again they are all available on Scotlandspeople. No sign of a birth for William Dick c 1795. There was one in Abernethy 17.10.1805 (father John but no mother’s name recorded).
I looked for John & Janet Dick in the 1841 Scottish census. There were 5 possible matches in Fife. You would need to pay to view them to see if any might be that family. (They might have died or moved away from Fife by 1841 of course).
The Fife marriage and the 3 childrens baptisms were all Church of Scotland (ie Presbyterian). Looking at the 1901 census of Co Tyrone, there were 24 people named Dick and all but one were also Presbyterian (which, in Ireland, normally points to Scots origins). It wasn’t a common surname and there were in fact only about 4 families in the county in 1901. Living in the townlands of Moyle, Douglasburn & Lislea. (Some of these are near Ardstraw which is where McCoy’s Dick military records relate to).
http://www.census.nationalarchives.ie/search/
In the absence of any other clues, you might want to search Presbyterian records in Tyrone for those townlands. The records are unlikely to be on-line, and you would need to get a researcher to go through them in PRONI in Belfast.
If you think your family were farmers in Ireland, then they are likely to be listed in the tithe applotment records. These are the only William & John Dicks in those records in Tyrone in the 1830s:
Dick, John-Tl: Churchtown Yr: 1834-Ardstraw-Tyrone
Dick, John-Tl: Knockanulla Yr: 1834-Ardstraw-Tyrone
Dick, William-T: Claggin Y: 1826-Derryloran-Tyrone
Dick, William-Tl: Knockanulla Yr: 1834-Ardstraw-Tyrone
Dick, Wm.-T: Lisnagleer Y: 1829-Pomeroy-Tyrone
T/Tl means townland. Again this gives you possible places to search. You might also want to go through the various Dick entries on this Co Tyrone website to see if any relate to your family:
https://cotyroneireland.com/index.html
