Sounds like 2 separate issues:
- it’s not powering off when you unplug it.
When you *unplug* the player, it should have a power saver time set & power itself off after a set time of inactivity. In Settings > System Settings > Power Saver, make sure you’re not set to “none”. Plugging your player in turns it on - nothing you can do about that - so use power saver to make sure it turns itself off.
- leaving the player plugged in to a computer that powers off.
I’ve had this happen with other Sansa products, so it’s probably normal for the Zip, too. I bought a simple <$5 USB wall charger (ebay or newegg.com) a long time ago to solve this problem with my laptop going into sleep mode and running down the Sansa battery overnight. Power saver doesn’t help in this case. You may have a cell phone or other charger that would work so check before purchasing a new wall charger. *Bonus* when the player is plugged into the wall or car, you can still use it (when it’s plugged into the computer, the data connection prevents it from working). Search the forum for “wall charger” and you’ll find specs.