Hello Ken and welcome to the site, firstly you do not specify the denomination, this is important as we rely on church records at this remove, I had assumed Church of Ireland given the name but a search on Roots Ireland does not return a David, there are 3 Stinson born in that period, 1829 plus or minus 5 years, two in Antrim and one in Limerick, I was surprised to find the Antrim ones were a C of I and an RC while the Limerick one is RC, if they were Presbyterian it is another story, Methodists I understand have not relaeased their records online.
Ashfield I know in later years, was the estate of the 3rd Earl of Leitrim, assassinated in Donegal circa 1878, he had land in Donegal; Leitrim; Galway and Cavan, he appears to have sold up Galway around 1860, I have ancestors who worked for him and I cannot find records circa 1850 but know at least one of his estates had a private church so not sure if they kept or released records, C of I records were lost in the civil war of 1921 in the public records office, about 50% or thereabouts so can be fruitless search there, RC records are patchy at that remove also, I am not sure if the 3rd Earls father was active at Ashfield but they later had yeomanry there and the later fellow was a recruiting officer for the Inniskilling Fusillers, when ever they came into being.
I also found in the Registry of Deeds project a number of Stinson records for land (land registry only got under way in the 1700s) you can see those records or a synopis here, a large tranche in Clare, also Tyrone and Armagh. https://irishdeedsindex.net/deeds_index/name_index.php It really only confirms the name in Ireland at that time, a search of the Down Survey does not show that they received land during the Plantation of Ireland in the mid 1600s.
Ashfield is an electoral district in Cavan of 17 townland and you can see it here, https://www.townlands.ie/cavan/ashfield/ and the parish appears to be Drumgoon, the town there is Cootehill after a Lord Coote, on this note I will touch the DNA of which I know little, it is possible that if the family came from England and were Church of Ireland (Protestant) and worked within the estate they married others of the same persuasion in Ireland, ie they lived worked; married etc for a few generations in Ireland but married within other families who came with hem or were already here and then left for Canada or the US. This is sort of what happened with the so called Scotch Irish.
I did a general search on Roots for Stinson born in Ireland from 1790 to 1810 and got 43, none in Cavan, variants were returned, some in nearby counties and some in Limerick, on the Deeds Registry they were in Clare, Limerick borders Clare in the South West of Ireland.
I have both RC and C of I ancestors in Cavan myself and the area is still fairly rural, did you search Griffiths by area rather than name, also some small bits of early census for Cavan were found but very small bits. It is on the census website just select year and county to see. I see 7 hits for Stinson in Griffiths Valuation for 1857 including a James in Cootehill, landlord is a Richey, never saw the name until last Monday night when Catherine Ryan a Canadian comedian working in London and on Channel 4 televison had a Richey ancestor from Donegal in Ireland.
Well Ken I said a lot but not got you any further on the road, there is a Cavan Genealogy centre that I have found reasonable cost wise but they may have some info or you could try the local history section of the Cavan library, a lot of areas have produced local history books in the recent past.
GoodLuck,
Pat
