Just to complicate matters unless you are blessed with great knee joints don't drop your cadence to climb if anything I tend to spin faster in an easier gear, Mashing/grinding up hills is probaly faster short term but I know it kills my knees.
I used to have the same hill climbing issue and hated hills now days I actually enjoy climbing hills it gives you a buzz to know how much you have improved, just keep logging the miles the hill climbing will come. I just came back from climbing the side of a mountain on a MTB for 2 hours, I enjoyed the ride up but the trip down whooooot. This from a person who used to drive to the bike track as I couldn't cope with the hills where I live.
Don't worry about trying to push the bigger ring just use whats required to get the job done, you will start using the higher gears soon enough.