Were the checks that your ex cashed in early 2022 refunds for your 2021 tax returns? Why were there two checks? Was one for federal and the other for state, or were they for two different years? If the tax refund checks were payable to both of you, you must have filed a joint tax return. But if your divorce was final in May 2021 you should not have filed a joint return for 2021. If you want to try to get your share of the tax refunds, you probably need to consult your divorce lawyer. If the amount is small, it might not be worth the trouble and expense.
If you did improperly file a joint tax return for 2021, even though you were divorced before the end of the year, you should correct the returns. But it's not as simple as just filing an amended return, because you are not normally allowed to amend from a joint return to individual returns. Your situation is an exception to that rule. The IRS calls it an "invalid joint election." But the procedure for correcting it is complicated and would require help from a tax professional. Also, your ex-husband would have to participate, which I sense might be difficult. And of course correcting the invalid joint election will get involved with the refunds that were already paid for 2021.
Did you file your 2022 tax return and claim your dependent child? If you e-filed and the return was accepted, then no one else claimed the child. If you filed by mail you will get a letter from the IRS if anyone else claimed the child. If you have not yet filed for 2022, go ahead and file, and claim the child as your dependent. If you e-file and it is accepted, then, again, no one else claimed the child. If your e-file is rejected because your dependent has already been claimed by someone else, you will have to file by mail. You and the other person will both get letters from the IRS, and you will have to send the IRS proof that you are the custodial parent.
Being the managing conservator has nothing to do with taxes. Custody for tax purposes is not determined by your divorce decree. Under the tax law the custodial parent is the parent that the child lived with for more nights during the year.
