Just an update.
I used a surplus power supply, that I'm guessing is from a computer. My project worked great; no power issues.
The power supply used is an Artesyn NFN40-760. I ran the RC servo and the common anode RGB LEDs off of the 5V output and ran the Arduino MEGA off the 12V output.
Next I was going to try a 12V choke on my 12V wall pack and see if that fixes my problem as well.