According to this /r/TodayILearned post, Churchill's GP wrote him this prescription for alcohol during his visit to America in 1932.
Was this a legitimate way to get around the ban? Did the average person also attempt to do this?
Not the standard source but Ken Burns did a fantastic documentary on Prohibition that actually addresses this.
Prescriptions were extremely commonly used to circumvent the ban. Often times entire families would get a prescription because the amount was limited to, if I recall correctly, a quart a month.
Follow-up: in Churchill's case, was it really necessary? As a visiting dignitary, wouldn't some rules be naturally relaxed?