You're correct. Form 1116 Schedule B will be available on 3/31/22. You're also correct that this form cannot be e-filed.
However, what you have paid for is the tax preparation software. There is no additional charge for e-filing either a federal or state return using TurboTax Online. It's an inconvenience, I'm sure, to mail your return. But the only cost will be the cost of mailing. Be sure to mail your return using Certified Mail so that you get confirmation that the return has been received. So sorry for the inconvenience.
Here are instructions for printing your return for mailing.
How do I print and mail my return in TurboTax Online?