No, it won't.
If you're sticking to the rules as written, then this sequence runs afoul of the Other Activity On Your Turn rule:
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
If you want to interact with a second object, you need to use your action.
You can't stow your weapon and then re-draw it in the same turn using free interactions because that's two interactions, and the second one would require using an action.
The only thing Dual Wielder changes is when you draw/sheathe one weapon, you can do the same with a second weapon at the same time, which isn't what you're trying to do here.