The stock rear banshee shocks, in all actuality are not BAD shocks. The reason they get a bad rap is because they needed to be valved from the factory for such a broad type of riding/rider weight, that most people don't find it very good.
The design of the shock is very similar to that of a works, or PEP shock. It's just valved and sprung for such a broad audience that it doesn't fit everyone perfectly.
I believe the "adjustment" you're talking about is the little flathead screw that's on the end cap of the reservoir. That is NOT an adjustment, and don't mess with it. Basically there are two ways to pressurize rebuildable shocks with nitrogen; With a shrader valve (just like on car tires), or by what's called a "needle fill valve", which is very similar to the needle you use to pump up a soccerball, football, etc.
Maybe you're talking about the two grooved rings at the top of the coil spring? The top ring is a "long ring" just like using another nut ontop of one to lock it into place. That bottom ring is called the "preload adjuster". Adjusting the preload will essentiall compress the spring more, and make it stiffer. Most people don't need to mess with the preload at all.
If you're interested in getting that rear shock dialed in to perfection, shoot me a PM. I've rebuilt and revalved quite a few members shocks on here and the stock rear shock with custom valving is a night and day difference.
EDIT: I just re-read your post agian. Simply put, NO. Running larger tires will not have any effect on your rear shock in any way.
Hope I helped.