Why not just try and find out what a simple change of the cassette gearing will do for you? Assuming your rear DR is not a short cage, then you might be able to put a 10 speed 11-32 or 11-34 on there and not have to change or adjust anything other than add back a few links to your chain or purchase a new chain with the cassette.

I just did the same with my son's 10 speed Tiagra taking it from 12-28 to 11-32. I used to be able to embarrass him on some hills, now I'm afraid those days are over.

11 speeds in itself is not going to do anything for the problem you are describing.