The reason why it needs empty slots for changing gearsets is because back in beta, if you have RingA and RingB, with two gearsets:

SetA: RingA on left hand, RingB on right hand
SetB: RingB on left hand, RingA on right hand

And you switch from the one set to the other, then only one ring would be equipped and the other finger will remain empty. This made the gearset function unreliable. So now when changing gearsets, it always first unequip one of the rings before starting to equip stuff in the target gearset---resulting in the problem that if your ring tab in the AC is full, the whole operation fails.

But even without this problem, we simply need more ring storage space in the AC. Having 2 ring tabs without specifically designating which corresponds to which finger would work.

I still want to suggest an overhaul of the Armory Chest. Make it a huge, big 300-slot storage, and use the tabs to just filter the gear for display. This will immediately solve a lot of the problems we have with AC storage because most of the times only one or two categories are full. And while you're at it, give us an "Expanded View" of the Armory Chest as well, please. If you must have tabs for the PlayStation 3/4 folks, making it Mainhand+Armor in one tab, Offhand+Accessories in the second tab would be worlds better than the current 4-inch display armory chest.