Jeep Wrangler shudders/stalls when put in gear after cold start

I have a 2012 Jeep Wrangler Unlimited auto trans, around 80,000 miles.

This started a few months ago. When I start it cold, like after sitting overnight, it starts fine, sounds fine.

But when I put it in gear (D or R), it either shudders and RPM’s drop and feels like it’s going to stall, or it actually stalls. It usually does a full stall when it’s really cold (like below 30 degrees). Otherwise, it does the shudder.

When it stalls, it starts right back up, and then it is OK – I can put it in gear and it acts normally.

The only other thing, potentially related, is that it seems to run a little rough when moving very slowly (like < 5 mph). But at idle it seems OK.

I changed the transmission fluid, and that made no difference.

Any ideas about what to look at?