Again, I am very sorry you are having trouble with SO2. I'm also sorry to hear you are not using iTunes to backup to a computer.
I'm having a hard time trying to explain what could be wrong if you haven't upgraded either the operating system, and/or SO2.
Just to verify we've tried all the simple things, have you closed all the applications and rebooted your device? (That sometimes works miracles.)
Your guess about getting close to your device memory limit might be a plausible explanation. I wonder if you have any applications or files (music, photos, books) taking up resources that you could delete or offload to free up memory, then re-try SO2 to see if your documents are visible. If freeing up memory allows SO2 to run and you find your documents are still present, you could copy them to one of the free online file storage services (Dropbox or Google Drive). (I'm not sure if iCloud for backup/sync is an option with your operating system.)
I recommend you try to free up memory to try to start SO2. If it launches, the next step would be to copy your files to a cloud based location. If neither of these steps works, we can try more intrusive measures.
As you are aware, upgrading your operating system is not without risk. You may have applications running that won't function under the latest operating system. Upgrading SO2 to current might be an option, but it is possible that doing so would over write your stored local files.