Dear Evan Hansen isn't eligible for Drama Desk this year. The nominating committee (and the voters) weren't invited to see it on Broadway. (Last year- when it WAS eligible - I know some voters may not have been able to get in to see it either).
I assume the same is true for Outer Critics.
Natasha, Pierre... is less clear. DD voters (and I assume Outer Critics) were invited to it. Not sure how many elements will be deemed eligible- would assume Groban and other new cast members, as well as the design elements. I would doubt it's eligible for Best Musical, but I'm not sure.