By US standards, both Boston and Washington -- certainly the parts that are likely to be on any first-time tourist's itinerary -- are very walkable. The public transport in those places is also good by US standards (though nowhere near the coverage and scope of the greater New York area, but there's nowhere else in the US like it).

In July the Northeast corridor is prone to be hot and humid, though it unsurprisingly gets worse the farther south you go. For that reason alone I'd probably pick Boston instead of Washington on this trip, but I have a particular loathing for hot humid weather. (Of course, the heat and humidity are worse in the South than in the Northeast at this time of year, but in most places in the South you are likely to spend less time walking and more time in an air-conditioned vehicle or space, which mitigates the heat and humidity somewhat).

I'd also say that if you like big cities and it's your first time in New York, you can absolutely spend 8 days there without running out of things to do. In particular, it would give you an opportunity to explore more farther-flung parts of the cities that first-time tourists (very understandably) tend to ignore and would let you come away with an understanding of the city that is more nuanced than the skyscrapers of midtown Manhattan.