Amended return, did not e-file but Filing instructions say state already has the amended return and not to send paper, but how did state get it?

Yes, there are only two ways they can get it, you e-file it, or you print and mail it. If you did neither of those, they don't have it yet. 

TurboTax assumes that if you go all the way to the end, you followed through and either submitted it for e-file, or printed and mailed it.