Dear Kierano
Welcome to Ireland Reaching Out and apologies for the delay in getting back to you.
I had a look on rootsireland.ie and there is no baptism record in Co. Galway for a William Douglas between 1793 and 1803. Given the early period of your search, this is not entirely surprising as parish records are only available for that period in very few cases.
One way around this is to explore the School Teacher information. You could try to access the Education Commissioners reports - some of these are available online at the website listed below:
http://www.dippam.ac.uk/eppi/documents/10935/page/248601
The reports refer to schools and their teachers and pupils. Only some of the reports are available online. As William seems to have been in the military in 1822 it might be difficult to find out much about him.
Some info about Schools' records is available from the National Archives, Dublin:
http://www.nationalarchives.ie/topics/Nat_Schools/natschs.html
It is likely that William was a teacher in a hedge or pay school - schools that were in operation prior to the official national school system that began in the 1830s.
If you learn anything new, let us know
Best wishes
Clare Doyle
Genealogy Support