Yes, you can do this
As explored in this question, it seems plausible that you can grapple an enemy and then drag/shove them into empty space so they fall.
If you successfully grapple an enemy using your first attack, you grab them and the following applies:
Moving a Grappled Creature. When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
So you can't move as quickly, but assuming there's a well within half your normal movement distance, you could get you and your opponent adjacent to the well. You can end a grapple whenever you like:
The condition specifies the things that end it, and you can release the target whenever you like (no action required).
So there is no action cost to simply release your grab if you've got the enemy in a precarious position; you can obviously grapple, drag, and release in a single turn. Many GMs would balk at you moving your enemy over an empty space and then dropping them, though. In that case, you could use a shove once you've got your enemy next to the drop in order to push them down it:
If you succeed, you either knock the target prone or push it 5 feet away from you.
That would require the use of another attack, but as you have the Extra Attack feature, you've got enough attacks available to do that in a single turn. Luckily, it's also explicitly clarified you can move between individual attacks you make as part of one Attack action with Extra Attack:
If you take an action that includes more than one weapon attack, you can break up your movement even further by moving between those attacks.
So, as long as you've got enough movement available to get to all the places you need to be, keeping in mind that your movement speed is probably halved while you're grappling unless your enemy is really Tiny, your Extra Attack feature allows you to both grab and shove your enemy in a single turn.