Frozen in place when talking to NPCs

I've got a really specific, really odd problem. It only started recently, when I was about to hit level 24 Archer.

Every time I talk to an NPC (seemingly excluding vendors), namely quest NPCs and the chocobokeeps, my character freezes for around five to ten minutes, sometimes not responding at all, before the dialogue finally appears. The same thing happens when I exit the dialogue window. My action bars disappear, I lose control of my character and just stand there for about ten minutes.

Everything else works in the game. Combat runs smoothly, I can craft, I can sort my inventory and look through all my logs, but I freeze every time I talk to an NPC.
The rest of the game has run completely smoothly up to this point with the exception of some mild connection lag around level 20. My internet connection is at a solid 40MB/s while playing this game, my processor is staying at a decent temperature and there's no strain on my computer.
This one problem is making the whole game unplayable, since I can't get any quests or hand anything in. I saw someone with the same problem on the PS3, so I doubt it's a hardware issue. I've already tried restarting the game (several times), restarting my router and reinstalling the client, but nothing seems to work.

Anyone else run into this problem or know how to fix it?