I amended and submitted a return, received a "SUCCESS!" response, then another screen showed "Your transmission didn't go through".

I'm getting this error on my amendment as well.  I'll be interested to see if anyone has any solutions!  

Still getting this message AFTER the "SUCCESS!" message: "A server problem occurred while processing your e-file. Please try to e-file again and if that fails contact customer service."

TurboTax is on their ass!

Funny thing - I can't get hold of "customer service". Their chatbot is useless.

Thank you for reporting this experience. We are aware and looking into it further.