You must release the target to attack, probably ending the grapple

D&D 5e has a broad intent that any word which is not explicitly given special rules meaning, instead has its plain English meaning, though I don't have a citation prepared for it. In plain English we see this:

Using at least one free hand, you try to seize the target ... you can release the target whenever you like

While "release" technically has multiple meanings, I find this context unambiguous in having Release mean "stop holding with your hand". From this plain English reading, I think it's clear that between the time that you "seize the target" and the time that you "release the target" your hand is occupied by that target; in other words, in order to use both your hands for a 2-handed weapon attack or other purpose you must release the target.

There does remain some ambiguity as to whether that actually ends the condition though. The sentence in question states:

The condition specifies the things that end it, and you can release the target whenever you like

But there isn't a strict connection indicating that "release the target" is an additional thing, not part of the set specified by the condition, which ends the grapple. It does seem likely that it was intended to be so based on the juxtaposition of the concepts, and supported by the subsequent section which unambiguously states that the means of ending a grapple are not entirely restricted to those specified by the condition.

Regarding "Common sense"

In a comment you mentioned that you don't bring real-world common sense into D&D, which is fine. Sometimes the rules explicitly contradict reality; after all you can't complain about someone casting a fireball on account of "magic isn't real". However, D&D also isn't a computer Rules Enforcement System where ambiguous edge cases have a single answer that's always correct and non-negotiable. Instead, D&D is adjudicated by Some Guy who, in my experience, is much more likely to be swayed by "It doesn't make sense that my Half-Orc Barbarian is physically incapable of throwing a gnome" than by "the rules don't say that letting go stops the grapple"; your mileage may vary depending on the DM. In general, I would say that ambiguous situations not clearly covered by rules are more likely to take whichever interpretation most closely matches common sense even if other times common sense is ignored.

No, this won't work exactly the way you describe.

Two-weapon fighting requires two weapons to activate

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.

The rules for Two-Weapon Fighting require you to be holding both weapons at the time you use it. It say you use a weapon that you are holding in the other hand. Not you will be holding. Two-Weapon Fighting requires you to be holding two weapons to use it. It isn't called One-weapon-and-maybe-another-later Fighting for a reason ;).

That means you cannot use Two-Weapon Fighting unless you are holding both daggers at the time of the attack.

This question talks more about this specific issue.

If you start with both daggers in hand it works

If you start with two daggers in hand the order would go like this:

(Which hand you use right/left, doesn't actually matter. 5e does not have the concept of an "offhand". So, you can flip right for left below easily.)

1. Attack action to do one dagger attack (right hand)
2. Drop dagger (right hand)
3. Grapple using Extra Attack (right hand)
4. Use bonus action to perform TWF attack with other dagger (left hand)

There is no reason you cannot take your bonus action TWF attack after your Extra Attack grapple because the rules say:

You choose when to take a bonus action during your turn, unless the bonus action’s timing is specified...

In fact, this is the default place for the TWF to go. So there's not much to debate here. Regardless, since the TWF attack is a bonus action and the timing is not specified, you can take it any time during your turn.¹

Also, since you started with two light melee weapons in your hands you meet the requirements for TWF. It doesn't matter that you drop one subsequently.

^{1 - It is unclear if you can technically insert the bonus action TWF attack between the 1st and 2nd attacks though according to Jeremy Crawford here. This isn't relevant to this case though. (thanks @V2Blast for pointing this out)}