Paper feed picks up second sheet before finishing the first

Clean the roller and try again.

David Lambert "The ECU controls paper motion in the printer by continuously monitoring the various paper sensors and coordinating the timing with the other print processes. For a detailed explanation of paper movement and the interaction of the photosensors and solenoid with the paper movement process, see “Printer paper-feed system” on page 67.

Motor

The motor is controlled by the ECU. The motor provides all of the printer paper movement.

Printer laser/scanner drive

The ECU sends signals to the laser/scanner assembly to modulate the laser diode “on” and “off” and to drive the laser/scanner motor"

So it could be the ECU. Use the SM LJ1200 for a deep dive

There are pickup roller and separation pad. You can remove both from printer -those are replaceable parts. There are 2 latches or whatever they call them - for roller to get free and you can use flat screwdriver to release separation pad. I would suggest - to clean or replace both (they should be cheap) and that could solve your problem. Also that could be metal piece that sticks to solenoid, when engages and not releasing after cycle of one sheet. Take cover off and check how solenoid is working. Good luck!