Drug store was common usage in the U.S. during the Civil War war. A quick way to see word usage in context, in different time periods, is a search like this, setting the dates to whatever you want:
Or for a survey of excerpts from various other period newspapers, many from the deep south, Vicki Betts' site is good. It can be searched by putting this search string into google: site:www.uttyler.edu/vbetts/ "drug store" The pages are large, so once it loads, use the "find" function to find the word or phrase on the page.
One could also search for "apothecary" the same way and compare the context in which the words were used. Or, for that matter, "pharmacy," or even "chemist," to see if the British name was used in the US.
This suggests "druggist" or "pharmacy" would be a more common usage. Against that, there's news items like this from the New-York Journal, October 29, 1772: