The 4-conductor cords that don't fit the remote are NOT handset cords. Handset cords have the RJ9 plug, the ones that don't fit are most likely wall cords with the RJ11 plug. Almost all the handset cords are coiled, so do a Google Search on telephone handset curly cord, and you'll find tons of choices. Just make sure you get the cord with all four wires, a few only had two wires as you have discovered.
