The US Soldier & Sailor System lists 9 William Douglas in the USCt. You can perhaps get the" right" William Douglas from their list.

You can find the link onder "Getting Started" on th etop menu-bar on the entry screen to Afrigeneas --

Then Choose Afrigeneas Links

Getting his exact regiment will be helpful when you try to sort through the 58 matches listed on Ancestry yo see if he has pension records.