Unarmed strikes with kicks and elbows and such aren’t just flavor, it’s written in the rules that you can use any part of your body.

Instead of using a weapon to make a melee attack, you can use a punch, kick, head-butt, or similar forceful blow. In game terms, this is an Unarmed Strike—a melee attack that involves you using your body to damage, grapple, or shove a target within 5 feet of you.

The mechanics don’t state you need a free hand anywhere.