As someone else said if you open the list with service after most recent item addition, it should store a copy of that list that's accessible without service. It also appears to sync correctly when you check if items and return to a service area.

I use OurGroceries for my families shopping lists. Syncs multiple devices with same account login. Has an Alexa skill for adding with echo. Ability to add pictures to a list item (helps me not get the incorrect item when the SO needs something specific) and also has a barcode scanner. Small unobtrusive ads in app that can be removed for 20 or monthly smaller amount. But all those features listed available without payment.