So, pretty bummed out here. On my second playthrough and just broke Esbern out of Riften. Fast traveled to Riverwood and there's no dialogue between them like I remember. They just stand right next to each other. Talk to Delphine and she scolds me for leaving Esbern behind when we're standing right next to him.

Tried going back to his house/room/cell. Tried asking him if he knows the way out which put him on auto pilot up until the room before the Thieves Guild tomb exit. When I talk to him it's wait here and follow me and a few other topics. Tried getting arrested, paralyzing and knocking out Esbern, pickpocketing Delphine. Pretty much everything I can think of to try and trigger it.

Out of ideas.

I'm not sure exactly what part of the quest you are on.... but if you are bringing Esbern to Delphine and her dialogue is not triggering.... maybe...

Check out the bugs listed on http://elderscrolls.wikia.com/wiki/Alduin%27s_Wall

particularly the ones about "When Esbern's conversation is skipped too fast, it might not trigger Delphine's dialogue... " and see if the fix listed there works for you.

Yeah that's not it man. I get inside the Inn and he's still following me around when I know he should be triggering something and walk straight up to her and chat, or at least that's what I remember from my first playthrough. He's stuck with a follow/wait command and not initiating dialogue. The game also starts to hick-up a lot like it's trying to force the scene but it won't go through.

If nothing on that page worked for you...

The only other thing I can suggest is to try deleting your game cache

"Clearing the Game Cache on the Xbox 360 is a separate function from clearing the System Cache. In order to do so, you must hold the LEFT Bumper, RIGHT Bumper, and the X button all at the same time on the Xbox 360 controller as a game is launching. Once you reach the Main Menu or Title Screen of the game, you can let go of the buttons. There will be no notification to you that this process has completed because it is all done passively."

Other things you can try:
maybe attack Esbern/Delphine then yield right away/pay off bounty.
Fus Ro Dah?

Tried 'em all, I think I'm SOL. Fooling around with unrelenting force I got him to say once "so this is where Delphine has been hiding..." or something along those lines and his wait/follow commands disappeared. However no conversation between them triggered after that and I couldn't recreate it.