Yes, a 19V @ 3.14A adapter will light up a series string as you describe. But not well, at all. I have similar 12V @ 5A LED strips, myself, and I can operate them at a lot less than 12V. But the strips won't operate at their rated output using that adapter.
If you want the full brightness, and you say you do, and if you want to run two of them from the same power supply then you will need to support 120W of power. This means a supply that is rated as better than 24V @ 5A or else better than 12V @ 10A. However, it's probably best to use 12V supply capable of more than 10A and to star the two strings to the power supply directly. Don't chain them, if possible, as it's likely that there will be voltage drops in the wiring at some length away.
If you locate your power supply in one place and the LED strings somewhere else (let's say, as lighting underneath a stairway railing) then you must consider factors such as voltage drop across the wiring between the two systems, heating that may occur in that wiring, heating that occurs in the power supply located in some enclosed space, tripping hazards, etc. It gets less and less fun as you work through all the necessary details, in cases like that.
I see another answer here cites a "per isolated meter" figure. But I don't gather you are operating these in isolated meter lengths. Just a note.