Why do my arms and hands keep falling asleep? Could this be something serious?

Yes. Your arms going to sleep could be due to a pinched nerve in your neck and may be treatable with some pt and meds. I would get it checked out with your doc.
Paresthesias. Tingling of the arms and hands can be something potentially serious. Compression in the cervical spine area from lifting, carrying bags, or injury can put pressure on the nerves radiating from the neck. The nerves beyond the pressure do not react normally and fire erratically causing the tingling or paresthesia. If it is short-lived/ resolves, monitor it. If it becomes more persistent md review.