Can I get 7A of current from two 3.5A rated batteries (without an increase in V)?
Technically yes.
Probably yes.
Possibly no.

Hard interconnection: The supplies will have diodes in the output so connecting foreign external voltage may be OK when it is operating and might be OK when it transitions from off to running. There is a possibility that the regulation / feedback does not like to see 19V on the output when it has not provided power and may not start.
Similarly, if one supply is set at say 19.1V out and the other at 18.9 V out,
Best case: the 18.9V supply sees 19.1V so concludes tha all is well so lays low. The 19.1V supply cannot supply the load and 'ags' until the 18.9V supply cuts in to take up the excess.
Worst case: The two supplies play 'pass the parcel' in some manner. the hv version is overloaded and start to shut down or completely shuts down. The LV version leaps in and is in turn overloaded. It shuts down and ...
But, it MAY "just work"

Series output isolating diodes:  A Schottky diode or FET super-diode is connected externally to each of the supplies and the two diodes are commoned. Having a shared capacitor at the common point probably helps. I'd expect this to work well, but nothing is 100% when two control systems interact and each is not 'aware' that the other is there. Size of cap may make a difference. Larger reduces swings when serttling down. Too large may inhibit startup.

Adding a diode with say 0.5V drop loses 0.5/19 ~~= 3% = tolerable. The Gigabyte 19V input supply can almost certaionly to;lerate lower voltages than this.