Story fits

I have a short story that is giving me fits. I started it the first time probably a year ago and then had to stop to write another. I had the story pretty much figured out, although there were certain details I couldn't decide on, and had a couple of ways I could go in. I finally sat down to finish the durn thing, got maybe 3/4 of the way in and realized I had a logic problem. Much time contemplating it and letting it percolate around. Then I wondered if I should cut out the problematic element and take the story back to its original direction. Finally after some discussion with a friend, decided I could work it out as it was--if I presented it in the right way to the reader. Went to write it down to "END" and another logic problem popped up. Damn, and double damn and a few other words came to mind. Then the other day on my walk, a perfect solution presented itself. Yay! I can finish it. It involves killing off one of my darlings, but it's ok. Worked on it yesterday, and guess what. Another logic problem with the new "solution". Now with the darling dead, who can my hero have a conversation with to release backstory details? Sheesh. So what do I do now? Revive the character? Or throw in a new one? The bad thing is I really like this world, its different enough that I think will appeal without driving off readers with its strangeness. Dropping the story is not an option. And crap, how much longer can I set it aside to percolate? Pray for me, dear readers. Send a lightning bolt my way.