Bike won't shift at all
I'm familiar with RSX brifters (brake and shift combo units)  If that's what you have then its likely the grease has gummed up, stopping the under-lever from engaging the release.

You might be able to finangle it a couple times by manipulating the underlever carefully, and you'll feel it catch.   A blast in the guts of the brifter with brake cleaner or similar can help short term too, but for the long term fix its at least an hour of messing around.

Here's a pre-existing answer where I wrote up the full procedure for cleaning a RSX brifter:

STI shifter sometimes doesn&#39;t catch in cooler weather

And here's the howto video showing... how to.

                
             

Answer: if a squirt of brake cleaner helps, then the full fix should make it right again.