I use ES File Explorer and Total Commander (each one is better for some things and worse for others). For moving files around, TC runs in split screen in landscape, so you can have the file in one pane and and the folder you want to move it to in the other pane. (Then just drag the file's icon and it'll ask if you want to move or copy the file. Since it's on the same medium, moving is safe.)
As far as the read only problem, as of Android 4, an app can only write to a folder it owns, unless that folder is marked writable by everyone, or the app is running as root. It you create a folder with, say, TC, TC can save a file to that folder.