My current setup is to run xmonad as my window manager within an otherwise normal Gnome session. Is there a way to do this with Sway, possibly with some other desktop environment if I had to?
No, this is not currently possible, but it's something we're working on. It requires buy-in from e.g. Gnome, which is hard to get. KDE is more amenable to this.
By the way, check out waymonad, it's based on the same wlroots library as sway:
By the way, check out waymonad, it's based on the same wlroots library as sway:
https://github.com/waymonad/waymonad