By the late-18th/early-19th century the idea of Atomism had been hashed over for literally centuries. They weren't going back to the Ancients at that point. Dalton's own discussions of Atomism were rooted more in the work of Newton (who made various corpuscular assumptions in his work) than they were in Democritus or Epicurus.

If you go back to the earlier discussions, e.g. Robert Boyle, you get a much more complicated picture. Boyle's Atomism was certainly done with knowledge of the Ancient philosophers on the subject, but his more immediate inspiration were discussions within alchemical theory about the nature of matter (which are more Aristotelian than Democritus/Epicurus; e.g. "what is matter made of that allows it to be X sometimes and Y another?").

The sort of place where I've seen explicit invocations of Democritus and Epicurus has been in discussions about the vacuum and Plenism (Atomism's antagonist philosophy, which argues for a continuum of matter, and lack of vacuum). The line between the philosophical and the empirical in this discussion, in the 17th century anyway (again, over a century before Dalton), is completely fuzzy. Hence one of the biggest arguments was between Boyle (a "chemist" in a modern framing) and Thomas Hobbes (a "philosopher"), who both saw themselves as doing semi-empirical, semi-philosophical work. Hobbes deployed Democritus and Epicurus to argue against Boyle's conception of the vacuum, and thus against Boyle's interpretation of his late 17th-century air-pump experiments. Which is just to say: everyone in that time would have recognized that these figures were involved in putting out philosophies of Atomism, but even by the 17th century it wasn't required to go back to them to get the ideas, there were plenty of other sources.

Dalton's accomplishment was not in re-inventing or even re-invoking Atomism. It was in trying to give it a strong physical basis and tie that into its possible chemical implications. It is very much the project of someone who has already stopped caring about what the Ancients said about such a matter, which itself is a very late-Enlightenment, post-chemical revolution sort of move. An earlier figure, like Boyle, cared a bit about the Ancients — not so much because he thought everything they said was accurate, but because they formed sort of a bedrock of understanding that he could push back against or build off of. And because he did think that ultimately what he was doing was a form of philosophy, even if he made an argument that new machines (like the air-pump) could serve as a run-around to philosophical quandaries. And the influence of alchemical theorizing on Boyle cannot be overstated — it was core to his understanding of what might have been going on in all of his experimental work.

This is a big topic and I don't claim to know every in and out of it (I am not an early modernist in the slightest, though I dip a toe into these readings sometimes for teaching purposes, and find the Boyle stuff pretty interesting for its strangeness). On Boyle v. Hobbes, Shapin and Schaffer's Leviathan and the Air-Pump (1985) is the classic text. On Dalton's Atomism, I found this article very useful a little while back: Alan J. Rocke, "In Search of El Dorado: John Dalton and the Origins of the Atomic Theory," Social Research 72, no. 1 (Spring 2005), 125-158. Rocke does a great job of situating Dalton's Newtonism and what exactly he was trying to accomplish. On Boyle's (and others') alchemical atomism (and various other Atomisms at the time), see esp. William R. Newman, "The Significance of 'Chymical Atomism,'" Early Science and Medicine 14, no. 1/3 (2009), 248-264. Newman is one of the great scholars of Boyle and alchemy. It is a difficult world to translate into modern physical/chemical terms on the whole — that is just not how they saw it working, and you sort of have to dive in on their own terms.