Maybe Midnight Commander? It can open FTP-connections (and shell connections, too, allowing secure copying of files over SSH rather than FTP), has that well-known dual pane layout, and, while not a GUI application, can be controlled the same way a GUI application can without even needing X. :)

IIRC GNOME has FTP and SFTP support, so you should be able to just use Nautilus.

I'm not using GNOME. I'm using XFCE, and unfortunately in my experience Thunar (the XFCE file manager) doesn't handle those sorts of connections too well. I think it's more to do with the FUSE layer beneath it (possibly the same driver that's used by GNOME).

For sftp, you can just mount it locally with sshfs and then use $FILE_MANAGER

Unfortunately, like I say, I've tried that (using Thunar - the otherwise fantastic file manager in XFCE). I'm not sure if it's Thunar's fault, or it's the FUSE layer beneath it, but I've always found it to be a little unreliable. Plus I like to be able to see connection logs and manage the activity queue in the same way I do with FileZilla.

unfortunately I haven't. For the most part, I'm a Windows user, so I've been using Explorer for plain FTP to my phone (which, while not great, does do the job more-than-well-enough for my uses)

Here's a recommendation: stop using FTP.