Nancy, Records in the 1700's & 1800's show family members names spelled Birdwell, Burdwell, and sometimes Berdwell. While most eventually settled on the BIRDwell spelling, the BURDwell spelling is also still out there.
I've also wondered about your ancestor Joseph Burdwell. It's likely he is from our same family, though that is not certain, nor have I found anything connecting him to a known Birdwell family. I'd love to exchange info with you regarding his possible origins prior to Texas.