Hi MSS312,
welcome to the Apple (user-to-user) Coimmunity!
Re: "I have a large Apple account balance due to unused gift cards.
I am trying to use my balance and pay the remaining amount on a card. It keeps falling."
Please be aware Gift cards need to be used in their country of purchase.
While the all-new Apple Gift Card can be used for everything: Apple: products, accessories, services, and more, some earlier gift-cards need to be used for particular purchases (Eg: App Store, iTunes, or Products):
See: What type of gift card do I have? - Apple Support
and
What you can buy with your Apple Gift Card or Apple Account balance - Apple Support
NB: Some items may not be available for purchase with Gift Cards / Apple Account Balance:
You may want to check which payment methods Apple can accept in your country or region:
Payment methods that you can use with your Apple ID - Apple Support
You can often get expert shopping help during purchase of a product in the online store, by clicking to chat with a specialist.
Apple provides detailed, written "Shopping help" on many topics online: Help - Apple
Eg:
This is the Payments and pricing page: Payment, Financing, Refunds and VAT - Shopping Help - Apple
Eg: says:
"You can use up to eight Apple Gift Cards at a time when you purchase online from Apple, or if you’re placing an order by calling 1‑800‑MY‑APPLE. You can use your gift card(s) with an Apple Account Balance and Apple Pay or one credit card".
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Re: "I have checked the credit card info and called my bank.
It appears that the charges aren’t even going to the credit card".
If your gift cards can be used ok for the purchase you are trying to make:
You may need to check with your bank whether International purchases can be made with your card.
If you are purchasing an item to be shipped, seems Apple requires that the address listed with your card issuer needs to be the same as the shipping address.
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Re: "I called and spoke to customer service but they aren’t emailing anything despite promises of a solution
in 24 hours".
Guessing it is Apple customer service you called, and not your bank ?
You may be able to get up to date information online if you received a case ID .
See: How to find your Case or Repair ID - Apple Support foir more details.
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More Help or Support:
You may want to online chat / phone / email with Apple's Billing and Subscriptions Support Team:
There's a "Get Support" link at the bottom of their website: Subscriptions and Billing - Official Apple Support
See a list of Apple phone numbers around the world : Contact Apple for support and service - Apple Support
All the best :-)