You can use other verbs with the phrase. Go is the most common, but you can also quit cold turkey, or kick something cold turkey. There may be others.
As to the phrase's origin, Etymonline favors the "quick preparation" theory and indicates there was a period of time where it was not associated with kicking a bad habit. It also curiously Cf.'s cold shoulder:
"without preparation," 1910; narrower sense of "withdrawal from an addictive substance" (originally heroin) first recorded 1921. Cold turkey is a food that requires little preparation, so "to quit like cold turkey" is to do so suddenly and without preparation. Cf. cold shoulder.
Here's the entry on cold shoulder:
1816, in the figurative sense of "icy reception," first in Sir Walter Scott, probably originally a literal figure, but commonly used with a punning reference to "cold shoulder of mutton," considered a poor man's dish and thus, perhaps, something one would set out for an unwanted guest with deliberate intention to convey displeasure.
How often have we admired the poor knight, who, to avoid the snares of bribery and dependence, was found making a second dinner from a cold shoulder of mutton, above the most affluent courtier, who had sold himself to others for a splendid pension! ["No Fiction," 1820]
I'll do a search for first usages.
Found the 1910 reference from The Trail of '98 by Robert William Service, though it's not clear to me how exactly the phrase is being used in this passage:
Couldn't find any reference before this. I'll keep looking for first drug reference.