Hi Alison,
I love my Paperwhite...but yes, some have had issues with the screen. I think they are in the minority but what do i know
As for your old one, I've sold a bunch of them. There is an option under the settings to "restore to factory defaults" and once you de-register it from you Amazon account, then all your Amazon purchased books would be removed anyway.
I usually de-register the Kindle, then restore to factory defaults. This wipes the Kindle and takes it all the way back to the welcome screen.
Even if you don't restore it, definitely de-register it. Otherwise, the one that gets it could be one-click buying books for you!