Oil Spots on Laundry
Could it be the oily stains are just not visible on light clothing, or the loads are run at different temperatures, or some other variation in way of explanation for the different results? Otherwise it implies there is something about your dark clothing that is the source, which I find difficult to believe.

I really think the source is the machine itself. We once had a well used machine that started doing the exact same thing. It happened to all clothing. We never really determined the cause. We eventually gave up trying to stop the stains. As soon as we used a different machine, we never had this kind of staining again.

The only thing I can imagine is oil from the machine's mechanicals somehow found its way into the wash water. This could possibly be through a perished seal at the pump, or maybe solenoid valves, or maybe it was an external source, once introduced difficult to get rid of. If we could be sure of the source, a replacement part could solve the problem. But I'm just guessing and am not confident that replacing some parts would solve the problem.

Unless you hear of plausible fix, I would consider replacing the machine straight away. Blindly replacing parts gets expensive, that money could better go towards a replacement machine. Washers are not inexpensive, but neither is clothing. It doesn't take a lot of ruined clothing to start costing more than a replacement machine. Do consider any other plausible fixes first, this is the action of last resort.

I'm sure it seems extreme to suggest a complete replacement based on a brief Internet exchange, but I've been in your position. The thought of how many clothes we ruined still sickens me. I would never suggest this if it hadn't happened to me personally.