Can you just walk up and purchase generic tickets or do you have to order them ahead of time?

You can definitely buy generics at any of the Event Registration desks and at Customer Service.
Maybe also at Will Call (if you're in line anyway for your other stuff) and hall HQs.

Also, the e-ticket app is supposed to let you purchase an e-ticket right at the event (so you don't need to carry enough generics for True Dungeon).