Yes, that sequence works
Player’s Handbook, page 190:
You can also interact with one object or feature of the environment for free, during either your move ar 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.
So yes, you could draw and attack with the first short sword on round 1, as per the rule above. Round 2, you use your action to attack with your sword and your free interaction to draw your second sword. As you are now holding a second weapon in your other hand, and you have just made an attack with a light weapon, the rule for Two-Weapon Fighting, PHB page 195, comes into play:
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to atlack with a different light melee weapon that you're holding in the other hand.
Rules as fun
Whilst the Rules as Written would say you can only interact with one object for free either during your movement or your action, and interacting with another object would require you to use your action, the DM could rule that, on round 1, you can draw both weapons despite not having the feat.
Mike Mearls said on Twitter that “the DM is free to make a call, based on the situation” in reference to being only allowed strictly one free interaction.
Note that this may somewhat undermine the features of the Dual Wielder feat. However, as the feat does give other benefits besides being allowed to draw two weapons - benefits you would not get unless you took the feat - the DM may allow for a player to draw to weapons in one turn, simply to “avoid punishing players for that stuff by charging an action”, in Mike Mearl’s own words.