Welcome, Laura!
Unfortunately it's not possible to write ADFs to disk without an actual Amiga or without other specialized hardware (Catweasel, Kyroflux, etc.). While the Amiga can read and write PC-format 720K disks (with CrossDOS), the reverse isn't true. You can't write ADFs to disk with just a standard USB floppy drive.
So I think your best bet is to try to get Amiga Explorer working. Then you'll be able to write ADFs over the serial connection using the 500's floppy drive. I've only ever used Amiga Explorer over TCP/IP, so I don't have any advice on using it over Serial/USB, but it sounds like you're on the right track with getting the new cable. Is the USB adapter definitely working? Cloanto has some decent support documentation on their site about making the hardware connection.
One thing that jumps out at me is that your WinUAE version is fairly old, which makes me wonder if your version of Amiga Explorer might also be out-of-date. Make sure you've got the latest to take advantage of bugfixes. While it's not likely, there's also the possibility that the 500's serial port is fried. (Just in case you forgot: never hotplug anything into an Amiga.)
What are your plans for the system? i.e., are you just looking to play games or are you looking to set up an actual computing environment to do other things as well? The answer to that question will influence recommendations for upgrade paths.