To apply the weird new shuffling rule.
I'm actually getting better at remembering this. It does help me not mess up and forget how many cards I've already drawn from Smithy, etc.
To exchange Travellers
I do this constantly. And forget Scheme. And Treasury. And probably Alchemists, though one guy in my group dislikes Alchemy, so I don't get to play it often.
To leave newly played Duration cards out during Cleanup
OK, there's an easy solution for this. Just play fresh Durations sideways, above your other played cards. At the start of your turn, you resolve each Duration card, turning the ones you're done with right-side-up and moving them down to where you'll play your other cards. Then you just have to remember not to discard sideways cards during Clean-up. It's really foolproof once you get in the habit. I haven't messed up a Duration in years.
If I were smart, I'd do something similar for Travellers, etc.