SIMPSONVILLE, S.C.—On a recent Saturday, Edward Johnson hurried into Harvard’s Liquor & Wine and made a beeline toward a 4-foot-wide section of bourbon. There he bypassed the Old Crow and Jim Beam and reached up to grab one of four stubby bottles of Blanton’s, a rye bourbon. At the register, the cashier had a 10-year-old bottle of Henry McKenna Single Barrel waiting for him. It was set aside that morning.

Mr. Johnson shelled out $107 for the bourbon and then headed to a nearby Sam’s Club, where he spent about $100 on diapers for his 19-month-old daughter. People observing this routine might wonder if he has a drinking problem. He doesn’t. What he has is a serious case of anxiety.

It began two years ago, when Mr. Johnson and his brother-in-law heard of an alleged bourbon shortage. He has been making monthly liquor and diaper runs ever since.

“It scared us and fear is a motivating factor,” said Mr. Johnson, a 36-year-old managing partner of Old Colony Furniture, who now has a stash of about 50 bottles of bourbon. The collection, he says, grew out of an urge to make sure there would never be a time “I wanted bourbon and couldn’t get it.” After a while, he says, the habit “took on a life of its own.”

He isn’t alone. Panic has gripped bourbon enthusiasts across the country, and they are amassing stockpiles of it, hoping to guard against shortages and price hikes.