It sounds rather as though you go charging at the hills and regard it as failure to have to go down through the gears. Believe me when I say I've been there and done that. My next mistake was to drop down to bottom gear way too soon, causing me to run out of energy through spinning too fast for too long. Getting it right can take some practice and repetition of your local hill climbs, trying different gears and trying to get a little further each time before you have to put a foot down for a rest. Cycling really does use different muscles than running does, and it will take a period of adjustment. I have a marathon running friend who has never been able to keep up with me on bike rides, while i get the stitch just running for a bus.
In my case I found that the best single thing I could do to improve my climbing was to relax my body position (straight back) to make deep breathing easier, and to change gear only as and when breathing became more laboured. If you stay in the right gear to keep your breathing under control it soon all gets easier, and you will find that climbing starts to become easier. Riding smart can also help ..... smaller hills can be easier if you sprint at them, letting your momentum do some of the work, but being ready to drop down through the gears at exactly the right moment. You will start to get a feel for this.
I have gone from getting out of breath riding over a motorway bridge 12 years ago to bagging a few first category Tour de France climbs (even though I'm a 300 pounder!), so I am testament to the fact that you can and will find a way. Good luck to you.