CA DMV has same requirements..Are you stating that your driver's license is only valid for operating a motorcycle? I didn't know that was an option - being issued one only valid for motorcycles. Here is The Great State of Washington you're first issued a DL for motor vehicles, then you can apply and be tested for a rider for motorcycles.
Following up on LEDphile's suggestion:To the original question, easiest option for moving a small amount of stuff without a car/van/truck is probably something like the Uhaul U-Box or the PODS shipping containers
My neighbor's girlfriend used a PODS to mover her entire apartment full of stuff. I'd forgotten about that. She had 30 days to unload it too (or rather he did, she just told him where to put it).To the original question, easiest option for moving a small amount of stuff without a car/van/truck is probably something like the Uhaul U-Box or the PODS shipping containers