Yes, you will enter the 2023 Form 1099-R with code P and J on your 2023 return since you had taxes withheld. Enter it exactly as you received it. The 2023 code P will not do anything to the 2023 tax return income, but the withholdings will be applied to 2023 (line 25b of Form 1040).
No, you do not need to amend your 2022 tax return if you had already entered the Form 1099-R with code P and J on your 2022 tax return.
You should have a 2022 Form 8606 which will show the basis of $6,000 from the nondeductible traditional IRA contribution for 2022. Your second Form 1099-R is reporting the distribution from the Traditional IRA which was converted to a Roth IRA. This will have to be entered on your 2023 return as it was received.
You will also enter a non-deductible Traditional IRA contribution in your return for the 2023 contribution that was later converted to the Roth IRA.
Take a look at the following TurboTax help article for detailed instructions to follow, as well as what to do if you did not have the 2022 non-deductible contribution reported on your Form 8606:
How do I enter a backdoor Roth IRA conversion?
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