The Watson sons both James 1802-1851 and Andrew 1813-1867 were Handloom Woollen Weavers
William 1809-1881 was a Founder / Iron moulder.
I think the surname Rutherford might be significant to the Watson family, and worth investigating. There was a son John
Rutherford Watson born 1804; and the oldest son James Watson who married Ann Baird in 1832, named their first daughter
Rutherford b.1835.
So far I have not found a plausible connection for Watson and Rutherford but hopefully someone else may see something I am missing. Of course the name may come through the Duncan side of the family. You would need to explore those names a bit more.

Incidentally, you may find this link useful
https://www.scan.org.uk/researchrtools/lieutenancy.htmI have not looked in detail, but a quick look under Kelso comes up with #125 William Watson wright with Neil & brother, and also #123 Robert Watson.