My iPhone 4s won't pair with my PC, and iCloud hasn't completed a successful backup in over a month! What can I do?

Hi guys, I would really appreciate some help with this. I've been to my local Apple Store's Genius bar twice and they have told me that they will replace the iPhone with a brand new 4s, however there is nothing they can do to help me get my important data off my phone (and until I do I can't give them this phone).

I normally would be able to just use iCloud backup to get my photos, videos and dozens of important audio memos synced off my phone, however (after connecting to a wifi but not paired) my iPhone's iCloud app waits about an hour saying "Backing up... Estimating time remaining..." then keeps spitting out "iCloud Backup failed. Please try again later".

The other reason this is so frustrating is that my battery will ONLY charge when the phone is turned completely off - meaning, I only get to use my phone for about 1/2 the day before I need to turn my phone off to let it recharge. Plus the battery seems to drain more quickly than it used to. Anyways, it's a bit of a first-world nightmare, accept that the audio memos I want to get off are so large (<100MB) and I have so many of them that I can't get them off fast enough using Google Drive or Dropbox...the wifi connection speed gets me about one or two every other day. 😠

Any suggestions you might have will be really really appreciated...the fine folks at the Genius bar haven't been able to help, and Lord Google has been no help either. Please help me get my data off my iPhone - thank you!!

P.S. Also, I think it's a very odd coincidence that when I started using Evernote to record my audio memos (and have them synced off immediately), Evernote stopped syncing after about a dozen successful synced notes and now no longer syncs at all....sidenote: I'm scared to uninstall and reinstall the app for fear of losing the unsynced audio memos.