We stopped at Lowell MA to visit the New England Quilt Museum (photography prohibited) and the Boott Cotton Mills Museum (photography encouraged). The NPS, among others, managed to salvage the buildings and restore some of the machinery, to the extent that one room on one floor of one building has some running cotton mills:
A bit more detail:
The original mills used water power, as did much of New England’s industry, but moments after Watt worked the bugs out of that newfangled steam engine, water power was history. The museum uses a huge old electric motor, mounted on the ceiling, to drive the line shafts above the mills; the vibration shakes the entire building and they hand out ear plugs at the door, despite having only half a dozen mills operating at any time. The working environment, horrific though it was, attracted employees (largely young women) from across the region; it was a better deal than they had on the family farm.
Employees were, of course, prohibited from using cotton to plug their ears…
They sell the cloth in the museum shop and we’ll eventually have some kitchen towels.
The Smell of Molten Projects in the Morning 
I once heard, many moons ago, that these places were calle ‘mills’ beause initially they were powered, as you say, by water wheels, and the only similar things, at the time, were grain grinding ‘mills’, so anything using motive power came to be called a ‘mill’, as in papermill.
Another offshoot of this was that the people who maintained the machinery in these places were called ‘millwrights’, the same as the specialist ‘wheelwrights’ and ‘shipwrights’.
Each building constituted a vertically integrated factory that consumed raw cotton and produced finished cloth, so on the first floor they had ranks of cotton gins to process the bales. I only recently (re-) discovered that “gin” stands for “engine”, which makes perfect sense.
The cotton went through successive refinement as it proceeded upward through the building to the mills on the top floor. They discovered, after the fact, that all the shuttles slamming back and forth more-or-less synchronously in a myriad mills tended to shake the building: a Very Bad Idea in a brick structure. Later buildings reversed the organization, putting the gins on the top floor and the mills on the bottom.