Change my monthly plan to bill annually

There isn't an option to do that. You may want to make the suggestion to Apple via Feedback - iCloud - Apple

I have seen some suggestions here to buy an iTunes gift card online and redeem it to the account so that the charges will be debited to the gift card rather than your normal payment method.

You can't. Apple doesn't offer an annual option.

Old “mobile me” users had this luxury before the transition to iCloud many years ago. One time annual payment methods are now obsolete and Apple has no such payment available.

Sorry it wasn't better news.