The “I/Me” rule is very easy to get correct in written form, but I’ll admit, in conversation I do catch myself sometimes saying the wrong thing.
Davis’s has always been the correct possessive. It’s Davis’ that is the interloper.
Sing / sing poss / plural / plural poss
David / David’s / Davids / Davids’
Davis / Davis’s / Davises / Davises’
Jones / Jones’s / Joneses / Joneses’
Heinz / Heinz’s / Heinzes / Heinzes’
Congress / Congress’s / Congresses / Congresses’
There have been traditional exceptions for “classical” names like Jesus and Achilles, but I see no good reason for exceptions.
It really doesn’t matter as long as you are consistent, but even then it’s not that important.