You were correct to suspect that the Cheng Shin tires are the main culprit. Cheap tires rarely have any sort of puncture protection belt built into the them.

You should be very safe with the Marathon Plus tires that you ordered, and the price you paid is a bargain. It will take you some time to wear these out, but while riding them, pay attention to how you like the ride. As you gain experience, you may not need all the protection that a Marathon Plus affords, and there may be other ride characteristics that become desirable to you.