I can't belive I'm asking this, but; can I supply the Arduino with an external adapter of say 1 ampere, and then connect something to the Arduino that draws 1A, or do I need to wire a separate power supply to that device only, and then share grounds?
Separate!
You can use the same supply but don't supply power from the Arduino, supply it directly from the power supply. And do connect the grounds.
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The Rugged Motor Driver: two H-bridges, more power than an L298, fully protected
Thank you RC!