There is a marriage recorded on http://www.familysearch.org of an Alexander Baird and Elizabeth Rae 7 Aug 1817 ALLOA, Clackmannan, Scotland and 2 Aug 1817 Polmont, Stirling, Scotland. These would be the publication of the Marriage Banns in both parishes of the bride and groom. This was the usual way of proclaiming marriages in Scotland in the pre-1855 Statutory Registration days.

I don't know whether this is your couple or not, but you mentioned Alloway in Ayrshire, which might easily be confused with Alloa in Clackmannan, unless you are sure about the former, and that Alexander and Elizabeth actually came from Ayrshire and nowhere else. Unfortunately, I could find no records for any children, but this is not uncommon in the pre-1855 era, even similarly with marriages.

You could try looking at the original OPR (Old Parish Records) images on www.ScotlandsPeople.gov.uk but often very little more information (e.g. parents' names) is given.

I don't know what more to suggest at present unless anything in the above strikes a bell. The Archives of Ontario hold all the later records after their arrival in Canada. The 1901 Census can be useful if any of their children lived to that date as it gives the date of birth (and sometimes place) and the year of immigration.

Best of luck, Hugh Gemmell