Like what jgwilliams said, it sounds like you may still have the lever shifted into "2nd" gear (I.E. the large chainring position). This would cause it to not be able to complete a second upshift.

Your high limit screw doesn't move the derailleur outboard, it can only limit the outboard distance. This is why it didn't do anything when you adjusted it out. You would also need to increase the cable tension in order to move the derailleur further out. And in that regard, I have found that 11 speed Shimano FD's require a very high cable tension to function properly, like a REALLY high cable tension compared to older setups.

I would say follow the step by step instructions at si.shimano.com for your FD-6800 and it should be fine. If you still can't get it going then a FD adjust at your LBS shouldn't cost too much.