Leave them dripping, but better yet, keep them wrapped or warm AND keep them dripping.
The purpose for letting them drip is that the pipes underground are likely still above freezing, and the water itself will keep the pipes warm enough to not freeze.
if indoor and an outer wall, keep cabinet doors open to the room, to let warm air in, and even place a light in the cabinet, to stave off the cold.
if outdoors, wrap heavily in styrofoam and rags while it is above freezing, and or while they are dripping and still not frozen
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Keep them protected from freezing weather as well as high temperatures.
Either way you loose. Save money buy and your plumbing by covering them with utility tape or tubes.
Good luck and stop the worring by doing the right thing and sealing the problem.
even dripping they will freeze up
Leave the water running at a fast drip or trickle. Also leave the cabinet doors open at the sinks. When the weather permits, buy some pipe insulation and wrap all exposed pipes. That being done, if it gets cold enough, you will still need to leave the water running.
heated socks and tinfoil
I was always taught to leave it slowly dripping AND to leave the cupboard doors open so that the central heating in the house can get to the pipes as well.