Control of 12V stepper motor using L293D dual H-bridge
You wrote: "In my past experience, not all GND pins need to be connected to ground".

This is a very, very bad assumption. You are dealing with a switched high-current inductive load. You will get sharp voltage spikes during switching. GND is very critical. Take a look at the layout example from the TI datasheet below. See the crazy number of GND vias recommended for pins 4, 5, 12 and 13? And yet you have several not even connected.

These pins are probably GND for each push/pull output. By failing to GND these pins, you are forcing the GND current to go through a longer, higher impedance internal path and you are probably getting severe voltage spikes on the internal IC GND which can then cause the IC to latch up, and/or inject spikes into your digital 5V rail.