That instructable has a few key steps missing but I saw your message on the original (https://www.instructables.com/id/Polargraph-Drawing-Machine/) too.
This problem is normally where the motors draw so much current that there isn't enough left to run the Arduino, and it resets. However, having a split power supply is the way to solve this, and it sounds like you have this already. It's described here: https://learn.adafruit.com/adafruit-motor-shield/power-requirements
Let's just confirm my assumptions: The Arduino has USB running to it, and nothing in the power jack?
The motorshield has the jumper removed, and has a 5v 2.5A DC power supply wired into the screw terminals on it?
Run the test with the external power supply turned off (just unplug it from the wall). What happens?
Run the test with the motorshield removed completely. What happens?
You do have a big mismatch between your motors and your drivers, even with doubled L293Ds.
Your motor will draw 1.7 amps. Each L293D will supply up to 0.6 amps, so your piggy-backed ones will be able to supply at most 1.2 amps. Because there's two motors, the power supply will need to supply 2.4 amps, which it can.
The L293Ds can be pushed beyond the 0.6A spec by adding extra cooling - heatsinks and fans etc. Piggy-backing them makes this more important, since each one limits the ability of the previous one to keep cool.
If both motors draw maximum current, then your PSU needs to supply 3.4A. Because you have the bottleneck of the thermal capacity of the L293Ds in the middle, that's probably not happening - but if you fix that, it might!
So the strange thing about this though, is that the electrical power for the motors is completely separate to the electrical power for the arduino, and the issue you report is the arduino resetting. In theory, the motors could burn out completely (or have no power at all!) and the arduino should just keep on going.
My guess is that there is something overheating and causing a short circuit - though I admit I can't really imagine how.
The best (expensive) solution is to make sure the components are matched more closely: Motors that draw 600mA or less, with single L293Ds. If you need more power (like more torque from the motors I mean), use a higher voltage power supply. I used to use 400mA motors driven at 7.5v and it was plenty enough.
The next best solution is to load the motor drivers with heatsinks and blow fans on them - that might makes a difference. On the other hand, if you feel the chips now and find they aren't hot, then it probably isn't a thermal issue.
Each time you do something that's a bit unusual (piggy-backing, over-specced motors, under-specced power supply) you add an extra variable that can go wrong that it makes it tricky to understand the failure modes, because they interact with each other!