Shatner is Canadian (as was Doohan). FWIW.
Also, there is no single "American accent". There are hundreds of them.
As for toning it down: Nimoy was from Boston, so he would have had to tone it down a bit (you wouldn't really want a Vulcan speaking with a thick Boston accent

). But De Kelley kept his Southern accent, and most of the other actors kept theirs as well.
Doohan, OTOH, ran through like a dozen possible accents before he picked Scottish. And who knows how Walter Koenig came up with his (imho, overly broad) Russian accent...