Okay, we have what the book says, what Mearls said and how you're likely to want to play this in your game.
For the book, draw or Storage can be done for free, you can do one of these in turn as part of a movement or action. However, if you want to do both, you are stuck burning your action to do it. If you're TWF, you're in an even worse shape (although the Dual Wielder feat corrects the economy of action here).
PHB page p190 states:
You can also interact with an object or characteristic of the environment for free, either during your movement or your action.
Mearls has said in tweets that you agree to use your free action to exchange weapons instead of spending your action to draw and save. That the intention here was to limit excessive use and also do not burn in the economy of action to exchange weapons. (watch this convo)
Ultimately, for your game, I think it's up to you to decide between these two rules. Personally, I prefer my character to do not Stay burned by the economy of action here and have ruled that quick exchanges of weapons (especially when cargo exits are commonly used) can be done as part of the attack. I do not stink about that, and I assume that the PCs of my games are competent in what they do.
If you really to have to do this job, dripping Your weapon has no action, so you could do that. It is complicated and prone to problems if you move from your space (you can not raise your weapon again), but maybe that is the cost of juggling weapons?