I got the solenoid engine working tonight. I filmed a video of it in action, starting out around 5.5v and increasing to 7.5v. This has a dramatic effect on the engine’s RPM.
The solenoids only got a little warm during the filming of the video, so that’s a good sign. Next steps will be to diagram the circuit in EasyEDA and get a PCB made, then mount the whole thing on a pinewood derby car body. But this is good for now.