Can I span 11 feet 7 inches with doubled 2x6 deck joists?
Doubled 2x6 joists are probably strong enough, but they're not going to be rigid enough to prevent significant bounce. That's why joists spanning 12 feet are usually 2x10 or 2x12. You need to upgrade your joists and possibly the single remaining beam to eliminate the intermediate beam. 

Alternatively, consider cutting the intermediate beam into the joists to be flush on top, and add joist hangers. This would raise it by 5-1/2".