I withdrew money from a 529 in Dec 2021 but paid tuition in 2022. Is there anything I can do to avoid paying the taxes for my mistake?

Going forward, have the funds sent directly to the school to avoid this error. 

Yes, if you took the distribution in December 2021 and paid for classes for the first three months of 2022, it could be viewed that the intend of the 529 account is intact. It would be the same as if you returned the funds within 60 days and then took another distribution. Keep good records since it is possible that the IRS might question you about this. 

In TurboTax, you can select that it was used for education purposes HOWEVER next year you will need to allocate that distribution to an expense and not use that expense again on your 2022 tax year return. 

In other words, if you only have Tuition and Fees, you'll need to report that "tax-free assistance not reported on a tax form" as received in 2022 (or just lower the amount in Box 1 of the 1098-T).  If the distribution was used for Room and Board or other expenses that are not eligible to be used for a credit, there won't be much you need to do, but you can't use Room and Board as an expense for a different distribution. 

You didn't indicate if the student is yourself or a dependent, but there are options with allocating distributions, expenses and scholarships in order to be eligible for an education credit. You might want to read over IRS Pub 970 for clarity. It is not that difficult to read and has lots of good advice and great examples of how the credits work.

Also, since school years and tax years overlap, you need to keep and eye on the 1098-T as well as the student's school account statements to make sure you are matching events to the proper year. Keep copies with your tax file.  

 Pub 970