This is probably a stupid question, but how do I successfully flee from enemies? Everytime I disengage combat by using the "Run" command, the enemies re-engage me on the very next turn no matter what I do.

Well, I imagine you know that you and your enemies have a position on the battlefield. The best way to run away is either running to the map exit, or sometimes going around corners.
So yeah, they typically continue to chase after you, or shoot at you, until you're out of sight.

You will remain in combat as long as you or the enemy can see each other. The distance of engagement varies by enemy type.

I can't seem to even move a single step though, and it doesn't matter what direction I try to move. They re-engage me as soon as I hit any button.

What am I misunderstanding here?

Not necessarily. Ranged enemies who don't move a lot might allow you to get around a corner. Exits are really the most efficient way to run away, since enemies don't ever switch maps. Now, they'll probably be there when you get back, but hey, you're safe for the time being.

Oh sorry I noticed your other post just now.
As far as not being able to move a step, you need to choose run away, then it should show the map and allow you to push an arrow key. You won't actually SEE yourself move between squares, but you'll notice that the next time you run away, you did move.

Ak ok, I understand now. I was thinking that choosing run as the party command would allow me to instantly move after it showed the map screen. I didn't realize that I needed to still confirm the option and let that round play out. Thank you.