"Yeah, but - nobody in 1864 would have assumed morphine was addictive, since it hadn't been widely used by then. (First isolated in 1817, but its first large-scale use was in the American Civil War)."
According to Wikipedia, morphine was first isolated in 1804, first distributed in 1817, first commercially sold in 1827, and expanded sharply in use after the invention of the hypodermic needle in 1857.
American attempts to legally restrict the drug didn't start till 1875, with San Francisco's first (rather timid) anti-opium laws, but it seems reasonable to assume users would have begun to notice its addictive properties long before then.
By 1864, when the US Civil War had just one year to run, perhaps the common soldiers had seen enough of their wounded colleagues fall victim to the stuff to gossip among themselves that it seemed to be addictive. That said, I dare say there was a bit of creative hindsight in Clark's lyrics too.