Will upgrading to a paid account extend my free-tier period or credits?

No. Adding a credit card or moving to a paid account does not extend the 12-month free-tier period. That period is based on the date the account was created. Promotional credits are separate and have their own expiration date, which normally cannot be extended by changing the account type. Check the Billing Console’s credits section for the exact expiration date. Adding a payment method mainly allows charges for usage beyond the free-tier limits or after the free tier ends.

I wouldn’t expect upgrading to extend either the free tier or the credits. Use the remaining balance before its stated expiration date, and keep an eye on any services that may start generating charges once the free tier ends.