Re" Whenever I try to download an app from the App Store, it asks me to verify my billing info, even though I have updated my card
You may need to allow 24 hours for the update you made to come into effect everywhere it is used.
Check you are signed in (to your iPhone) with your correct Apple ID:
Sign in with your Apple ID - Apple Support
or
Check which Payment methods that you can use with your Apple ID - Apple Support in your country or region.
It may be the card-provider you need to verify with.
Eg: If you have disputed a charge from Apple, you may need to authorise payments to Apple again.
or
You may need to authorise international transactions before they can be made from your card.
or
If there is a problem from a previously unpaid purchase:
See: If a message says 'billing problem with previous purchase' or 'verification required' - Apple Support
More info:
What to do: If you can’t download or update apps on your iPhone or iPad - Apple Support
Billing and Subscriptions Support website (info / support / contact options etc):
Subscriptions and Billing - Official Apple Support