Yes, that is correct you cannot recharacterize a Roth conversion.
Yes, since you had pre-tax funds from your 401k rollover in your traditional IRA now the pro-rata rule applies. This means that with each distribution/ conversion, you will have a taxable and nontaxable part. You can see the remaining basis on line 14 of Form 8606, this basis can be carried forward. Therefore, each distribution/conversion in the future will have a taxable and nontaxable part until the basis is all used.
The Backdoor Roth only works if your traditional/SEP/SIMPLE IRAs are empty. For 2022 you won't be able to fix this and part of your $6,000 conversion will be taxable in 2022. But if you plan to use the backdoor Roth strategy in the future you might want to think about a reverse rollover where you rollover IRA money to a company plan, like a 401(k). Only pre-tax funds can be rolled from an IRA to a company plan. Therefore, you would isolate the basis and could start the Backdoor Roth procedure fresh. But it only works if your employer allows it, not all plans do.
Please see these articles for additional reading: How to FIX Backdoor Roth IRA Screw-ups and When the Reverse Makes Sense: Benefits of a Reverse Rollover.
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