This is of course a "thought experiment," since nothing could stop the Earth's rotation in its present continental form short of magical technology, and from what I've read, the sun will expand into a red giant long before the Earth is dragged naturally into a sun-locked position by tidal forces. In fact, this may happen even before the Earth and Moon lock together in a natural 47-day cycle.
But if one side of the Earth faced the sun all the time, would it become uninhabitable everywhere? The atmosphere would continue to circulate to some extent. Certainly water on the dark side (which, unlike the Moon term, would really BE a "dark side") would freeze solid. Would the twilight zone be habitable? The polar areas would see "seasons" due to the tilt.
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