Available in a paper form  and in an electronic format are not the same thing. Looks like it should be ready on 3/8 if there are no more set backs.   If you need to file today then do it without the credit and amend later.   https://ttlc.intuit.com/questions/1908854-forms-availability-table-for-turbotax-federal-products-exc...

TT does not create or submit the form the IRS does.   The IRS creates an electronic version of the form called a schema which is a data protocol that tells TT how to encrypt the from data so the IRS computer can decrypt it.  The IRS has not done that and keeps pushing the estimated date out.

The "form" you see is simply a look alike of the paper version for the form that is only used for your data entry.  e-file "forms" never get transmitted to the IRS, only the encrypted data that you entered gets sent to the IRS and must be in the format specified by the IRS or the IRS computer could not read it.

each week the date changes when will it really be ready.

Can I file by mailing in forms printed out by Turbotax, and include form 8936 in that way?

You could mail in the return IF the program correctly calculates the return BUT  mailed in returns take 6 - 8 weeks to process  AND  you let a data input operator on the other end have the opportunity to miss enter your info ... it is best to wait and efile it.

And the printed form in TurboTax will have a "Do Not File" printed on it, so the IRS might just return it to you.     Be sure to check that the tax calculations are correct also - TurboTax's accuracy guarantee will be void if any form has "Do Not File" on it.