before removing the ferrite bead and was able to read 20.1V, but after removing it I get 0v? Any idea why that might be?
It probably "sparked" (shorted) one to many times, and it could be dead.
(An internal fues may have blown, but those those things often have to be cut-apart and epoxied or Bondoed back together... Not really recommended...)
Once i get the AC adapter back up and running I was planning on using a 7812 to provide 12v to some devices and then splitting that off with a 7805 for all the 5v devices. Will that work? Does that sound like a realistic plan? Thanks!
If you can get voltage out of it, Yes... Maybe... The 78xx series are linear regulators rated for about 1 Amp maximum. And since they are linear regulators, the more voltage you drop across them, the hotter they get (at the same current). That means you have to watch the power dissipation (Voltage x current = power) as well as the current (you can overheat it at less than it's maximum current rating), and you may need a heatsink.