Hi Renamay: I replied eatlier but my reply went to the wrong person!
I found some Archibalds but they were Loyalists and I think if yours was in North Carolina in 1812 he would have been in your army and therefore not a Loyalist. So perhaps you should look up some of your own country's military Regiments at that time. Also if your Archibald was b there in N. Carolina, the father had to be there at least nine months before, leaving that time nowhere near Loyalist time.
So that is well past that time. Try to determine names of his children. Most Scots used a Scottish "naming pattern"
which probably varied with their location in Scotland.
Do you know the names of his children? They often names every generation with the repetition of names, which has made it very difficult to trace family members.

Did he have a church association? Were his children baptized, etc? Are there any signatures on any papers or
names and places on a will or land transaction? It seems there has to be some item such as a Muster Roll of a Regiment which you could obtain from a source from or through genealogy access through your Government or Military sources.I would think he surely had to give his place of birth somewhere along the way.
Many families split loyalties and went their opposite ways.
I think that happened in mine and I am out ob a limb looking for somebody to which to attach my Charles. He may have been b in Inverness-shire, Scotland but the maybe in Quebec. There were family members on the Plains of Abraham in 1759 and many stayed in Quebec. I am still searching though at least I have everything since that time . I hope you reach your goal. Good luck ! Delphine