I have his son being Timothy Dunnagan. 1715 - 1752. He did leave Ireland and then lived in Tennesee.
Williams wife was Sarah Philips 1710 - 1750.
It seems that he was born and died in Limerick, Ireland. But lived in the US at some time because of Sarah. Sarah died in the US. That might be why he returned to Ireland.
I have William being born 1690, but I don't know when he died.