Most of my web development happens on a web server that I access over SSH - so instead of using an FTP client to log into my files and access them that way, instead I "mount" my web server to my computer over the network, and as far as my computer is concerned it appears in the file manager the same way a USB drive or SD card would: as removable media.
Once you have mounted your remote file system to your computer, it will appear in the Open and Save dialogs in every application, and it's easy to drag files between folders in your file manager on the remote drive as naturally as it is to work with files on your computer.
So do this I've used:
It's worth looking into as an idea, instead of shopping for another FTP client :D