I have had your Joseph Wills on my ( find him ) list for quite a while now and may now be hot on his tail. As you mentioned he claimed to have arrived in Adelaide in 1844. I have found a Joseph Wills who could be the son of the Joseph Wills you a searching that married Eliza Welchman in Adelaide in 1884 ( Cert no. 1154 )His fathers name was Joseph Wills and his mother's maiden name was Baker.
A search further through the South Australian marriage records for a Wills - Baker marriage should provide more informaton for you. Sorry I don't have those records.
Hoping that this info now puts you on the right track.