Try tweaking the alignment of the cage a little. It's supposed to be parallel to the chain ring of course but that can be difficult to actually eyeball accurately. If it's just a little out of alignment so the tailing end is a little bit angled inside (toward the frame), it might affect the ability to shift to the large ring. If it does appear aligned correctly to you, still turn it a bit so the tailing end angles out just a bit more. That might solve your problem. After that, you might have to adjust your limit screws again to avoid rubbing chain on cage.
This has helped me in the past. It was either that my eyes were not serving me well in the original alignment, or that the cage simply needed to be angled out a bit to work well. I don't know which was the issue, but the solution was to angle it out more than it was when it wasn't working.