Question

Any Scotland birth rec for Catherine Gilmour-788 Templeton, c1775-1783, mother's name Carswell?

Answer 1

A death memorial with image on Find-a-Grave.  Find A Grave Memorial indicates she was born about 1781, and died 21 June 1857 in Kilmarnock, Ayrshire.I also found an entry at Scotland's People for:  Elizabeth a potential sibling.

I hope that helps give you a place to start looking.