...I'll 2nd, 3rd, and 4th what everyone else has already told you...."spray the hell out of it with WD40".
I had the EXACT same problem with the left RSX shifter on my '96 Bianchi a few months ago, and took the advice of an apparent "expert" on them (Andy Zeigler
andyzeigler@gmail.com ) that I stumbled across on an old bike forum thread, and he advised me to spray lube it - and it worked!
If you still have problems with it, Andy says he rebuilds them for $25 ea. He says he's sucessfully ("sucessfully" being the operative word here) rebuilt several dozens of these old RSX's. You certianly can't get even cheap replacements for less than that.
I'm holding off til the end of the "season", before shipping my off to him for a little TLC tune-up.