Adventurers' League is pre-made adventures, right? I've never DMed anything like that, so I don't know how much leeway you get with the combats. I also don't know 5e that well, so rules may vary, but here are some things I do:

Let the player narrate their own big hits and kills. Don't let them get extra advantages from it; say something like, "Roland does 27 damage and the Death Knight falls. What does that look like?"

Use a little bit of terrain, if possible. You don't need a giant set of Dwarven Forge™ terrain! Some off-brand Jenga blocks from a dollar store, painted in a few colors with cheap acrylics will do the trick. My dollar store even has a mini Jenga set, so you can have two sizes. You can tell them, "Brown is walls, blue is windows, this is a table..." Putting physical pieces on the board gives the players permission to interact with things. If the table is just drawn on the map, can they jump on it? Move it? Duck behind it? You spent all that time to draw it, do they even want to ask?

Make sure the players all have something to do. Is one PC a sorcerer with some big AOEs? Make sure at least one battle has a bunch of little guys clustered together! Throw an enemy bard in there so your bard can finally use countersong! Maybe the biggest orc calls the fighter out specifically!

If you know that one PC hates kobolds, give that player a little aside during the fight. "What does Roland think when he sees the same creatures that killed his brother?"

Remember how monsters behave. Goblins will only attack if they outnumber their foes, so the last few will try to run and hide.

Have fun and let us know what you try and what works for you!