Given the option, I go shopping on my own, but that doesn’t happen very often. I find that writing the grocery list geographically (so that it matches the store layout) helps me get in and out of the store quickly, hopefully before meltdowns start.
The other thing that helps is to let Lily add to the list in-store – so long as she agrees to let me glance at it from time to time to see what we still need to buy!
The biggest help? Making one big shopping trip once a week (or less), and then staying away from the stores!