Daubeney seems to have been a typical English spelling from an early date, as far as one can guess from Latin records. Mostly it was translated as de albineio or de albiniaco, but those forms don't necessarily indicate the true origins of the families, which they probably forgot.
Dugdale called them de Albini in his Baronage. But he seems to have made that up, or got it from somebody else who made it up. This became tradition in that sector and was followed by Nicolas, Courthope and Cokayne in their Peerages.
Browning changed it to d'Albini, probably so he didn't get a lot of letters.
In Complete Peerage 2nd edition, Gibbs decided it was time to ditch Dugdale, and switched to d'Aubigny for the early families. I haven't found d'Aubigny in any original sources. It seems to be a reconstruction in Anglicized modern French of a hypothetical name that Daubeney is supposed to be a corruption of. But it's become totally conventional for the Arundel and Belvoir families.
Elias and his descendants are generally called Daubeney by English writers, following the spelling on Sir Giles's peerage patent, but Americans like to correct it to d'Aubigny.
As regards the Dabney line, Hugh
https://www.wikitree.com/genealogy/D'Aubigney-Descendants-14
and the next 5 generations seem to be taken from this Visitation (with a William skipped)
https://archive.org/stream/visitacionievisi32ryew#page/98/
which doesn't add much information. This village history
https://www.sharrington.org.uk/history/
doesn't add much more.
But that line is probably garbled. Blomefield's version
https://www.british-history.ac.uk/topographical-hist-norfolk/vol9/pp436-438
starts the Daubeneys with William, but has Broughtons to match the John and Robert at the top of the chart.
Whether this family had anything to do with any immigrants is a different question.