You can wield & attack with two weapons without any special class feature or feat.
Like most contests, however, some restrictions may apply. From the PHB, p195 (emphasis mine):
Two-Weapon Fighting
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 attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
As long as both weapons are light weapons, you can attack with both at the cost of an action for the first attack, as normal, and then a bonus action for the second attack. The second attack does not receive your ability modifier as a bonus to damage the way attacks normally would.
Fighters & some other classes have an option for their Fighting Style class feature which is also called, confusingly, Two-Weapon Fighting. This feature is not required to attack with two weapons as described above; the benefit it gives is to allow you to add your ability modifier to the second attack's damage.
The Dual Wielder feat is likewise not required to attack with two weapons as described above; it instead provides additional benefits and removes other restrictions/penalties when you choose to do so.
Dual Wielder and the Two-Weapon Fighting style make you better at fighting with a weapon in each hand, but are not required to do so.
No
But the argument is not so straightforward.
The problem with this question is that the RAW are murky enough to allow for some leeway in interpretation.
Going by RAI, the intention was clearly to use Two-Weapon Fighting with one-handed weapons, and not game the system by attacking with a versatile weapon while holding it in two hands, arguing that it still counts as a one-handed weapon for the Dual Wielder feat.
The Rules
Dual Wielder
You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light.
While this qualifies the longsword for two-weapon fighting, the Versatile property states that:
Versatile
This weapon can be used with one or two hands.
A damage value in parentheses appears with the property—the damage when the weapon is used with two hands to make a melee attack.
This explicitly states the need to use both hands to make a two-handed attack and use the bigger weapon dice, thus disqualifying this use case from the Dual Wielder requirement (one-handed melee weapons).
The Conclusion
While you can use a thrown handaxe and then attack with the longsword, you can only do so using the one-handed dice the Versatile weapon tag provides.
Best Answer
Yes
At least by the rules.
The rules for manipulating objects come under:
Other Activity on Your Turn
The rules for interacting with objects are limited to one object without using your action. As you are drawing and sheathing the same sword, the rules as written do not prevent you from doing it.
The fact that you are sheathing the sword at a different time than drawing it are not covered by the D&D rules as being important.
There is no second object here. The object interaction rules talk about how many objects you can interact with, not when you can interact with the object (other than restricting it to be during your action or move).
However
It is a common interpretation that the "one object" restriction actually means "one object interaction", not "you can interact with one object".
This is such a common interpretation, you are going to have people say it to you as gospel.
This relies on reading "interact" as a description of a singular discrete event, not as a description of your relationship with the object during the action or move.
So you should talk to your DM.