I've only been at this guitar thing for a few months, so others will hopefully have some good advise to help you. However I can tell you this my fingers aren't short but they are big. If I try to play an open A the traditional way I can only fit two fingers between the 1st and second fret. I can actually play the open A with two fingers but don't. I just have to put my index finger behind and between the second and third to make a good clean A. Sometimes you might have to use an alternate fingering.

What helped most, when I was having the same issue was slowing down and checking a string at a time. I think it's Lisa McCormick but it might be Caren that has the lessons on how to practice that talks about making micro-adjustments. At any rate what really helped me was working on just very slowly and cleanly setting my fingers down then checking string by string what was hitting that shouldn't and made the micro-adjustments mentioned in the video. Now I can often feel the adjacent string if I'm touching it and make the adjustments on the fly.

At one point I was convinced my hands were just to big to play the guitar. Now I'm thinking it'll be an advantage in the long run I can comfortably cover frets 1-5 without shifting my hand position, can go clear to the 7th with a stretch. And for those that are thinking it yes I play the bass ;)