Can you get naegleria fowleri from picking your nose after washing your hands, but your hands are dry?

No, never: You can only get this very rare infection by swimming in contaminated water. That usually means warm, often stagnant water in small lakes like swimming holes. In the US, most cases occur in the south during the summer. In any case, this is an extremely rare problem: in the 52 years from 1962 to 2013, there were only 132 reported cases in the US, an average of a little more than 2 cases per year.

Not a mechanism.: Between 1962 to 2016 there have been NO case-reports of Primary Amebic Meningoencephalitis caused by Naegleria fowleri occuring in the state of New York.