Actually you don't even need the printout for the Immigration, because they can see in the system that this passport holds a DTV. But it's never wrong to have a piece of paper... just in cases the phone die or whatever.
Another thing... since there seems to be some confusion in the last days... the idea behind the eVisa system is, that everything will be done online. There is no need to visit the Embassy or the Consulate. And also they don't want to see you there anymore... ;-)
Since nobody knows yet, how exatly the extension will work, yes... out and in is the best way at the moment. If the extension is really just 1,900 Baht and no additional paperwork is needed, then you can stay 360 days before you have to leave.