A couple more things to think about...

The skills coach I did a few sessions with used to remind us all the time that you need to have your butt perched right on the nose of the seat to have your weight forward for climbing. He would point to the "s" symbol on the nose of my seat and say "S is for Sphincter. Make sure that's where it is!"

Now I don't always agree that I need to be that far forward, but when you are going up steep hills and you need to make sure you manage to both maintain traction but not lift your front wheel, you need to move your weight forward (butt forward on nose of seat) and make sure you are not pulling up on the bars (make sure your elbows are low and remember that oft-quoted advice "boobs to the bars"). Experiment - you'll find your sweet spot for climbing.

The other thing was one I learned from a very old man. I used to climb in a mid-lowish gear and mentally when I changed down to my granny I expected to be able to maintain the same sort of momentum, so would spin so fast I would run out of puff and stall.

One day I was out with a friend and his 70yo father. The friend is a mountain goat and would leave us standing on the huge climbs. I would start off ok and then when I shifted down to granny, would pedal like a demon, run out of puff and resort to walking. The friend's dad would then ride slowly past me at just over walking pace. I would hop back on when I got my puff back, and then go past him, only to run out of steam some time later and have him pass me again.

Eventually I got sick of this, so one time, instead of passing him, I fell in behind him and tried to pedal slowly in granny like he did. It worked and I got up all the rest of the hills without walking.

I think I just had to modify my expectations about what would happen once I finally had to resort to granny. These hills were quite substantial (a few around the 20% grade and longish) so it didn't make sense to try to get up them the way I had been. Sometimes you just have to accept that a slow grind is going to get you there.

So my suggestion is to have that slightly bigger gear going as you were suggesting, drop back to granny on the hills, but settle into a cadence that suits you and keep your weight forward.

Oh - and one more thing another friend said that helped...

Attitude, Breathing, Rhythm.

You need to convince yourself you are going to get up this hill.
You need to breathe comfortably and rhythmically.
You need to adopt your own rhythm and pace. We all have one that is natural to us. Don't think you will do well if you try to adopt somebody else's when you are trying to do something that is physically difficult to you. Find your groove and stick to it for the hard stuff.