You can still attack, and use Extra Attack, but you can't use your 2-handed weapon to do it.
The Grapple Action specifies the following...
Using at least one free hand, you try to seize the target by making a grapple check instead of an attack roll [...] you can release the target whenever you like (no action required).
This pretty clearly states that you need at least 1 free hand to grapple your target, and that 'releasing them' ends the grapple. The problem comes that you're trying to use a two-handed weapon.
Two-Handed. This weapon requires two hands when you attack with it.
If you are Grappling a target, one of your hands is busy holding on to the person you have grappled. Therefore, you do not have two hands available to use a Two-handed weapon.
If you are using a one-handed weapon, you can still hit them with that. You may also choose to use Unarmed Attacks instead. Note that you do not have to put your Axe down to do this, because...
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
So you could grapple someone then headbutt them into submission, without putting your axe down.