Well, despite not getting any assistance from anyone on here, I have managed to get my machine setup the way I wanted it to be. The setup I was referring to earlier is actually called "Zaphod Mode", after the character Zaphod Beeblebrox, and needs to be declared in your Xorg.conf file.
Here is an example of my working configuration:
Identifier "Default Layout"
Screen 0 "Screen0" 0 0
Screen 1 "Screen1" LeftOf "Screen0"
Option "ZaphodHeads" "DVI-0"
Option "ZaphodHeads" "VGA-0"
As an aide to anyone else trying to setup up Zaphod Mode on their machines, the way I have it defined above will treat the Monitor connected to the DVI port as primary. If you want it to be the other way round, simply switch the "DVI-0" and "VGA-0" options in the screen sections. My configuration above also assumes that my second monitor is placed to the left of my primary. To changes this, simply change "LeftOf" to "RightOf" in the SeverLayout section. Also, make sure that driver specified in the Device sections matches that of the one for your graphics card.
Zaphod Mode has removed my problems when running full screen applications, and while I can no longer drag windows between screens, I am quite happy with how my system is setup. For me, having each monitor treated separately actually suits my workflow better, as it means I can simply ignore the other screen unless I actually feel that I want to use it. And for gamers there is simply no better option if you actually want to play your games full screen.