Hi Keith
Hope this might help. A privately printed book, "Records of a Family of Andersons of Peterhead from 1560" (1936) lists seven children of a William Anderson in Brora, Sutherland, and his wife Margaret Squire. One of them is Alexander who went to Ayrshire and "had a large family" - no further details given. However, the Clyne OPR confirms an "Alexander son of William Anderson and Margaret Square [sic] in Brora" baptised on 31 Dec 1787. I'm certain he's the man you are looking for and that he understated his age in both the 1841 and 1851 censuses (not unknown!) - perhaps to conceal the fact that he was so much older than his wife, or perhaps just vanity. There are trees on Ancestry (not that I ever trust them much) that give William's ancestry (correctly) for a few generations. I am descended from William's brother John.
The Andersons are a most interesting family and were the founders of the Orient Line of steamers.
I always wondered what happened to the Ayrshire branch and we assumed there were no male Andersons left now. It's a pleasure to know I'm obviously wrong.
I'm happy to exchange further info.
Best wishes
Pozzo