Do I need to strip off old wax coatings before applying a fresh wax?

This depends on the project and desired look. If looking to restore an old piece of furniture to make it look like new, removing old, dirty layers of wax with a Wax and Polish Remover is the best approach. If looking to retain the patina and natural character of a piece of wooden furniture, applying a fresh coat of wax will retain the character whilst restoring the look and feel of dry, worn, scuffed or scratched areas.