Alternate Wall Adapter?

Yes - providing they are DC output and their off-load voltage doesn't exceed say 14 volts.
Most cheap ones will fail the off-load limit : I have a 12 volt one which exceeds 20 volts off-load !!
Certainly the arduino will always present a load which will make an unstabilised wall-wart output drop to the arduino spec of 12 volts but without knowing by how much, it's safer to avoid anything that exceeds 14 volts.