Hi,
Welcome to RootsChat

Williams obituary says that he was born near Scarborough so I wonder if this could be him,
29 May 1796
Willerby by Scarborough, Yorkshire
William HALL
Mother: Ann
I can see an image of this baptism, it says,
May 29th 1796, William, son of Ann HALL of Staxton, a Bastard
I realise that this is a baptism and not a DOB but maybe he didn't know both dates. I think it would quite a coincidence to have two William HALLs using the same date in the same area.
Where do you find his mother's name as Elizabeth?
Regards,
Daisy