Apparently there is too much play, indicative of it being too worn to function properly. Would the RD-MC31 be compatible or is there a newer compatible set that I could use?
most shimano derailleurs will do, as long as you keep them "9-speed" and below. See also the biggest table here (the biggest table with the 1.7 derailleur ratio).
Make sure your replacement has at least the tooth capacity you have / you plan to use (big chainring + biggest sprocket wheel - smallest chainring - smallest sprocket wheel)
Do not replace the shifter with one for a different amount of gears, as that is the part that determines how far the derailleur will travel per gear change.