I don't recall any specific statement that the shuttle's on TOS were impulse-only. In fact, they are shown on screen as performing missions which would implicitly require the use of a warp drive or some sort of FTL system at any rate. I haven't actually watched enough of Enterprise
to know for sure, but i bet Archer's shuttlepods were only impulse. But we can be certain that the shuttles from TOS, TAS and all the movies were warp-capable. Exception: The wedge-shaped vulcan shuttle needed an attached warp sled, and was likely not warp-capable on it's own. Prior to ST5:TFF, it was assumed that the ship carried similar craft, but since then, I think e can assume that the refit always carried them too. And they look to have warp engines.