So I decided to start a new character and wanted to try out a dual blade wielding archer who uses Restoration and Illusion for support. Unfortunately, Bethesda made it impossible to go from dual wielding weapons then to spells and then back again without opening the favorites menu. I've tried searching the nexus for a mod that fixes this issue but I've had no luck. Does anyone know if there is a mod, or can it even be fixed at all? Is there a way to make spells auto-equip to your right hand instead of the left?

Thanks in advance for any answers you guys have. Don't think I can play a dual-wield character without some sort of fix.

The categorized favorites menu adds support for "equip sets". Not sure if this is exactly what you are looking for but who knows.

Here is a link to the mod. http://skyrim.nexusmods.com/mods/4862

I haven't tried with spells, but when I switch back to my main hand weapon from my bow, my offhand weapon gets re-equipped automatically.

Yeah, that happens with twohanders, so bows and 2h weapons. (maybe with ritual type 2H spells?)
I see it as one two handed slot which takes priority, and two one handed slots. So if you unequip the bow, it goes back to the one handed slots.
Unfortunately, those dual wielding spells or weapons don't get that option.
I haven't tried the equip sets, but it's probably your best bet.