You need to follow couple extra steps to make it happen after you migrate app to use Online Store 2.0.
Step 1: Navigate to Online store > Themes > Customize > Theme Settings > App Embeds > Enable Zip Code Validator > Save
Step 2. Go to app > Global Settings > Scroll to "App embed configuration" > Enable "Enable cart page warning as we're using only section validator". > Save
Please make sure to only enable this setting when using a section validator.
Once you save settings, you should see cart page warning without any issues. If you don't please contact our support with relevant information.