Everything you doing is fine I just don't understand why you'd have to open up a different ira to put in the rest of the 2022 contribution. You can leave and use the IRA you already have open for all contributions and conversions. All this will be combined on the tax return and the non deductible contributions you've made will be reflected in the form 8606 so you don't pay taxes on the money twice. In the TurboTax program, you'll need to enter in the non deductible traditional IRA contribution first, then go back and enter the 1099-r for the distribution/conversions that are reported to the IRS. By doing these entries out of order in the program you will have the correct information on the form 8606 when needed.
As always, before you file the return, you should review the entire return carefully. Especially pay attention to the form 8606 and make any corrections needed before you file instead of finding the mistake after you file and having to amend the return.
