From your description of the damage to the potatoes, you may be dealing with voles, a type of meadow mouse. Voles feed on plants and the roots of trees, shrubs, and  herbaceous plants. They create a surface runway system 1-2 inches wide with many burrows. They are difficult to control. You can use snap-type mouse traps baited with apples which are placed in the surface runaways or burrows. Some vegetable gardeners dig a trench around their potato bed and install a barrier of 24-inch hardware cloth. You can also place crushed oyster shells on the surface of the soil as they do not like to dig in this material.

Would I have better luck if the problem is voles if I grew the potatoes in burlap sacks or large planters?

Voles can chew through burlap sacks. You can try growing in bushel baskets or hard containers. Regardless you will need good drainage.
mh