Yes, you should drip faucets when it freezes hard, on any faucet fed by pipes in an outside wall, attic or unheated garage. A slow drip from both the hot and cold sides keeps water moving and, more important, gives pressure somewhere to go if ice forms. It is cheap insurance, not a cure.
The pressure relief matters more than the movement. The damage usually comes from water pressure trapped between the ice plug and a closed faucet, which is why pipes often fail on the thaw, as the guide on what to do if your pipes freeze explains. An open tap keeps that section from sealing. Moving water freezes more slowly, but a dripping line can still freeze in a long, windy cold snap.
Drip the faucets on outside walls, usually the kitchen sink and any vanity on an exterior wall, plus the fixture farthest from where water enters the house. Dallas Water Utilities advises letting hot and cold water drip slowly from inside faucets during a hard freeze and opening cabinet doors under sinks on outside walls. On a single-handle faucet, center the lever so both lines drip. Disconnect garden hoses before the freeze.
Dripping has limits. It does nothing for a pipe that is already frozen solid, and it does not protect an outdoor faucet or replace insulation on attic and garage lines. During the February 2021 storm, officials in Houston and Austin asked residents to stop running water or to conserve it because system pressure had dropped, so follow your water utility’s notice over any general tip. If the house will sit empty, turn the water off at the main water shut-off valve and drain the outside faucets instead.