Ok, I think I've figured it out. I used to be able to hit ESC while in SlickEdit, and in the command line (or whatever it's called) in the lower-left corner, enter: VS+new and a new instance would open. That no longer works, returning an Unknown Command error.
I am able to launch a second instance via a shortcut where I've added +new to the end of the command line. A little less convenient, but a workaround nonetheless.