For tires, I assume you could get ~6000 miles on a set, it depends on the tire and how you ride it. That is about the miles I got on a V-Strom 650. Personally, I would go with Continental Road Attack 3 tires. Those are expensive but awesome.
But even budget type tires today are really good. Hopefully you have a shop in the NH area that could help?
I grew up on the East Coast and I would say there is 0% chance you will have a rain free ride (full trip). And maybe even a 50% chance you will get rained on each day.
When we did a trip like you are planning, we altered the trip each day due to weather. If you roughly plan your trip down the mountain chains you should be able to either ride to the East or West up and over, etc based on fun/time/rain.
For example, the Blue Ridge Parkway, wonderful road ... but you might find on a particular day it is pouring rain on the BRP, but 10 miles west of there is dry.
As for fun roads, it you go up/over any of those mountain ranges you will be on fun roads.
Well, mostly.
We normally went with "camp when dry, hotel when wet" plan. The bugs can get pretty bad in June, warm muggy nights.
I would suggest sorting out your hotel around 3 pm. Like pull over somewhere, check maps/phone, figure out you will be in Soandso Town by your desired stopping time, look up hotels, call up and book a room. Then ride the next 2-3 hours to that town.
The wrong way to do that is to roll into a town at 7:30 pm hoping to find an empty room. Seems travelers (cars/families) start doing that around 4:30 pm or so and fill up the hotels. So you want to book your room before that. And doing it day by day allow you to adjust your route.
That is also assuming you are staying at some small town in the mountains. If you are on interstate and each stop has like 3-8 chain hotels then no worries with just rolling in at 7:30.
Oh, rain gear. Funny thing ... seems most buy it, packed in cases/bags ... riding along, get a few drops of rain, thinking it doesn't look so bad, keep going ... then a few more drops ... then by the time you think "this is bad, get the rain gear" you are already pretty wet!
That has only happened to me like 100 times.
The challenge with most rain gear is it doesn't allow moisture out ... so if it is a muggy 80-90 degree day with 70% chance of rain, what do you do? Start out with rain gear on? Kind of miserable. Or wait until it rains and try putting rain gear on, in the pouring rain, standing on the side of a rode in the weeds and mud.
Frogg Toggs has some reasonable priced rain gear, actually so many options it is a bit confusing.
I got Marmot stuff, not motorcycle specific, but reasonable price and I got the ones that do allow moisture out.
There are also plenty of moto specific, one or two piece rain gear options, that aren't too expensive (like $60-$120 range).
If you have mega dollars, their are fancy fancy outfits that you just wear every day that have vents and stuff and are rain proof so you just keep riding if it start raining. Talking KLIM or BMW type gear here. Too much money for me.
Hope things work out where you have a great trip!