Mid June you should be good. Most of the local schools are still in session.
Evan's Notch is a bit further of a drive (about 25-35mins from MT Washington) then what has been mentioned so far. They have 5 campgrounds plus some first come first served that is off RT13
Probably a typo, but ME113.. Don't know when you've been up there as of late, but last year Bull Brook FC/FS is closed. I have not seen a potential date of re-openning nor how they will. With it being cleared from logging, it makes another potential NF CG possible.
And yes, there are a few other spots, yet the pressure on those is going to be high...
I really like ME113 (Evans Notch) and have no issues with the NFCG's there. Theirs a new host at Wild River which I've heard conflicting comments about, so if you go there before me please post.
To the OP, Wild River is 5 odd miles down a dirt road. If you're looking for a night-over along ME113 and working Mt Washington, I suggest Hastings NFCG. Good host there (if still) and clean CG.