Careful with that - writing ANYTHING to a corrupt file system may corrupt it even more, beyond any chance of repair. I think your best bet is a full backup and reflash. Do you have any file manager installed other than the stock one? Can you open the terminal? The reason for asking is that you need to access files and directories above MyDocs, where the stock file manager can't reach.

If I were you, then depending on the extent of the damage, the only thing I would install - if not installed already - is the SSH server. Then connect to the phone from a PC via SSH and do a backup. Once done, you can try running fsck from BackupMenu but in my experience that is not going to go far, unless the damage is absolutely trivial. A better option is to connect the phone to a PC in BackupMenu mass storage mode (note this is different from "normal" mass storage mode as it exports two more partitions) and try to fsck your partitions. Or just back it up that way. You will need a Linux PC (at least a virtual one) for that.