I can see that you posted a query in the past, I know frustrating when information cannot be found, Donegal is regarded as one of the poorest counties for record holdings.
Below is a link with insights into researching Irish records with details on how parishes evolved.
https://irelandxo.com/ireland-xo/news/how-locate-parish-origin
The links below have visuals of Donegal parishes.
https://freepages.rootsweb.com/~bhilchey/genealogy/DonegalMain.html
http://donegalgenealogy.com/donparmap.htm
In Donegal appears most common spelling is McCullagh, it is a surname with many variants. The site below had 32 instances in County Donegal spread over 17 parishes. It is based on Griffiths valuation (1847-64), it did not return any with the spelling McCullough.
https://www.johngrenham.com/findasurname.php?surname=McCullagh
I searched again in the pay transcription site www.rootsireland.ie. 1813 is early for Donegal records, not all churches are transcribed, for some holdings only commence in the mid -1800 period. The only Hugh McCullough birth on the site for year 1813 was in the neighbouring county of Tyrone, as some distance from the border between the counties I believe an unlikely match. Within the county of Donegal there are no early records with the first name Hugn.
In regard to a DNA check where advised no success when used in the past, if it has been some time since used Ancestry.com site, I would suggest a revisit. I have been impressed that where found they break down origins by sub-areas within County Donegal. They may indicate that the surname is of Scottish origin with some limited detail as to where found in Ireland. As more people test all sites are improving their ability to locate connections.
Family Tree DNA reportedly has more people with Ulster roots than any other company and 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).
Marion Whelan - IrelandX
O - Ireland Reaching Out Volunteer.