Bdtyk

Pick a class that has access to the Extra Attack feature. For my example, I'll use my standby, Fighter McFighterson.

Let us say that McFighterson is 20th level, and possesses the Dual Wielder feat. He is currently holding a greatsword, and has two longswords accessible.

As a reminder, two weapon fighting says:

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...[snip]

And Dual Wielder:

[snip]... You can use two-weapon fighting even when the one handed melee weapons you are wielding aren’t light.

You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one. ...[snip]

Can McFighterson make all of the attacks granted by Extra Attack (3, in this case) with his greatsword, drop the greatsword "for free", draw two longswords (as permitted by Dual Wielder), use his last attack to attack with one longsword, then two-weapon-fight with the second longsword?

Making the attack sequence look something like:

Greatsword, Greatsword, Greatsword, Longsword1, Longsword2.

Related: Does Two-Weapon fighting work if you aren't holding both weapons when you take Attack action?

Yes, you may.

There is no limitation built into two-weapon fighting to deny that. It only requires taking the attack action, and making at least one melee attack with a one-handed weapon. Dropping the greatsword doesn't cost an action, drawing a weapon is considered interacting with an object, and drawing a second weapon is included in the object interaction thanks to the dual wielder feat. The PHB even specifies that drawing a weapon can be done in the middle of an action or movement, so drawing a weapon between extra attacks is permissible. As part of your attack action, you have attacked with a one handed weapon, which means you can use your bonus action for an off-handed attack.

Both weapons must be in hand for the whole sequence.

Per your own quote...

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

...one has to take the Attack action with the required weapons in each hand. Taking the Attack action triggers the bonus action swing with the different weapon. If one does not have both weapons in hand when one takes the Attack action, one hasn't met the criteria.

Dual Wielder removes the light restriction and allows drawing two weapons at once (saving on the interact-with-object part of the action economy)
, but it does not address or modify the "holding in one hand... different melee weapon... other hand" chain.

Extra Attack doesn't make a difference.

Extra Attack is a bit of a red herring. While it has the same trigger condition (taking the Attack action), it doesn't have the same restrictions. In fact, one could use two different weapons within that one Attack action (up to the total number of available attacks), and then do something completely different with one's bonus action.

Whether a character does (or does not) have Extra Attack doesn't change the requirements of two weapon fighting. Prior to gaining Extra Attack, one must have both weapons in hand. After gaining Extra Attack, one must still have both weapons in hand. If Extra Attack were meant to change the parameters of two weapon fighting, Extra Attack would say so.