Just store the list on your phone? It's better ux too. Unlike with pen and paper, you can check off items with a tap and rearrange them arbitrarily.
(It's a small phone and perfectly fits a very old camera case that holds some adapters in a pocket)
Any more than to pay. The advantage to paying with the phone is that through regular but not constant use, I am very unlikely to misplace it.
As noted, I sometimes do both. I think that my old shopping app fell by the wayside (as apps sometimes do) and I never replaced it.
Just store the list on your phone? It's better ux too. Unlike with pen and paper, you can check off items with a tap and rearrange them arbitrarily.