My first guess would be that you're not in the right gear. You shouldn't be pushing that hard. How steep are these hills and what kind of gearing do you have? I'm in Colorado, so I know climbs. I also ride a 'cross bike and don't have the widest range of gears. I would expect to hear you say you tire from spinning, not that your tired from pushing.
As for falling, if you're in the right gear and just need to build your climbing endurance, I'd put flat pedals on and do hill repeats until you build some stamina.
Keep at it! That's the only way it gets easier (once you have the shifting down!) It's common for beginners to get confused on shifting and not use the proper gearing for climbing.