There are two ways to fix this, one is a simple. The other one and would require going back to the drawing board on mechanics and animations.
Option 1: First Basic Attack
The first basic attack for every weapon in the game should allow you to do it while walking or running. That way, tapping basic attack every other second, can make you follow a person and whack him around without initiating a combo which slows or stops you from moving.
Option 2: Combos Take Movement in Mind
Have the combos for every weapon take account of your movement. No combo should prohibit or alter your movement in any way, this included stopping you or slowing you. This I believe would require a remake of all the animations when it comes to weapon combos and how it works.