If you were not able to use your entire Foreign Tax Credit, the remaining balance will be available for you to use in a future year as a carryover, as you mentioned.
TurboTax will carry the balance forward for you, so no need to redo your 2022 taxes. If you pay Foreign Tax every year on this stock, you will be able to continue to use the Foreign Tax Credit.
The maximum Foreign Tax Credit you receive is based on the amount of US Tax on the foreign income you received, which is why your credit may be reduced.
The Simplified Method is when you claim the FTC without filing form 1116. The penalty for doing this is you lose the right to claim any FTC carryover.
Here's more details on Claiming the Foreign Tax Credit.
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