I guess I should have tried this before, but I've narrowed down the behavior to one specific campaign/character. I started to notice some slight issues with it last night, but the issues today have been severe. It's not the longest campaign I have (I have one in Act 3), but it started to show after a significant Act 1 event and got worse this morning after Patch 3. The event in question is:
The Raid on the Grove, where I joined in on the side of Minthara and the Goblins
Things started slowing down after I completed the event last night, and they have been far worse today. I've checked all my other saves, and they don't have this issue, so something has been set that's causing the problem. I thought it might be:
The Idol of Silvanus, which I'd been carrying around
but it looks like that is not the case. I dropped it and went into another zone, and the issue persisted.