According to Urban Dictionary, a bottle run means:

To walk casually into the alcohol aisle of any grocery store and proceed to take a bottle, usually Jack Daniels, and walk out of the store like nothing happened. A variation can be done with a hand basket and several bottles. If one can handle the sketchiness of the act, a shopping cart and the fire exit can also be used. A bottle run will get you a good year on probation, so if someone behind your back tells you to stop- Run Forrest run!

But in Dotty's case it seems more likely that the term means taking empty bottles
back to the store for a small refund on each (usually after you've accumulated
enough of them to make it worth the trip), like they do in some states
in the U.S. as a means of encouraging recycling.
~hugs, Veronica