It would be better if the distribution (1099-Q) matched the 1098-T, but as long as you have documentation, you will be fine.
I think you (and your parents if they claim you) would be MUCH better to use the distribution for Room and Board.
Then, the tuition paid "out-of-pocket" may be used towards a credit.
It does NOT matter how the events transpired over the year, the IRS lets you allocate distributions, scholarships, and expenses however is best for the taxpayer. The IRS shows several great examples in Pub 970 of putting Distributions to Room and Board so that tuition can be used for a credit.
If you are eligible for an education credit, the IRS encourages you to take it.
In TurboTax, enter the 1099-Q first, then expenses including Room and Board.
You can pay for the first three months of the following year and claim it the year paid, so add the payment that crossed years. As long as you have proof that you made the distribution in 2022, you should be able to claim the expense for the January, February, and March classes.
Pub 970
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