@LLC3
First, you need to correct 2021, and you need to do this before you file your 2022 tax return. You may want to file a request for an extension for your 2022 return, in case you can't finish all the steps in time.
You need to prepare an amended return for 2021 to report the non-deductible IRA contribution. If the contribution was $6000 or less, then it won't change your tax owed, but you will generate a form 8606. You don't actually need to file the entire amended return, you just need to print, sign and mail the form 8606 (it is one of the forms that can sometimes be filed by itself). But the easiest way to prepare the form 8606 is to use the procedure to amend a tax return. Keep a copy for your records.
If you contributed more than $6000, you will owe a penalty for excessive contributions. You will have to check your records with the IRA custodian. It's too late to fix that, so you just have to pay the penalty. In that case, you would need to file the whole amended tax return, because it will include a form 8606 for the non-deductible contribution and a form 5329 for the excess contribution, and you will need to include a payment. You may be able to amend online and pay electronically. If you have to amend by mail, be sure to keep a copy, and use a mail service with tracking and proof of delivery.
Be aware that, if you made contributions in 2022 that were supposed to count for 2021, you had to tell the bank or broker that in advance. You may want to check with the broker to make sure your contributions were properly applied to 2021 and 2022 as you intended.
The next step is to file your 2022 tax return. You need to report the non-deductible IRA contribution. Turbotax will also ask for prior non-deductible amounts, which you get from the 2021 form 8606 (this is why you have to fix the 2021 return first, so you have the 2021 form 8606 when you prepare your 2022 return).
If you had an excess contribution in 2021, that will be penalized again (6%) in 2022 since those excess funds are still in your IRA. There is a way to fix that and avoid paying a duplicate penalty, but that will require an extra step on your part. After you fix your 2021 return, let us know if you had an excess contribution, and whether you want to pay the penalty again or remove it. But it has to be done before April 13 (to allow a few extra days for the bank to process it before April 18.)
By making non-deductible IRA contribution for 2022, your 2022 tax return will contain a new copy of form 8606 with updated information on your non-deductible IRA balances.
The third step is to rollover the IRA that contains the old 401(k) into your new 403B.
The fourth step is to do an IRA to Roth conversion of the non-deductible IRA funds. In this case, you would report this on your 2023 tax return, and you will need to refer to copies of your 2021 and 2022 form 8606's. The conversion will be non-taxable.
If you want to do another backdoor Roth conversion in 2023, you can do that after the conversion of the existing funds or you can do that at the same time.
