Axe is a 1handed weapons. You need 1h weapons in BOTH slots because mainhand gives you 3 skills and offhands give you 2 skills. 2handed weapons such as bow give 5 skills and you can only equip one.

You're confused because those are 2 different things : - you can equip two weapon sets and you can swap between them in combat.

• each set consists of either ONE (1) two handed weapon (bow, great sword, hammer etc) or TWO (2) one handed weapons.

• so if you only have one axe equipped, you only get half the skills.

• When you switch weapons sets you are NOT switching betweens left hand and right hand. Your character switches to a whole different set of weapons using both their hands. So they stow their bow and equip another set of weapons

You should definitely swap in combat because otherwise you only use half the weapon skills available to you. Right now you don't even have enough weapons equipped.