I would say you are keeping it way too wet. Take it out of the pot, wash the soil off the roots, see if there are any white ones (= growing roots),cut off all the branches and repot it, using lots of gravel or perilite mixed in the soil. Stop watering it!

Even if you do the above, its branches indicate its roots are rotted to me, in which case, it's a goner.

I agree with Kay. Never had any luck with rubber trees. Try growing a Schefflera. They are easy to grow and beautiful.

You can pick the plant out of the pot, set it on newspapers to absorb any excess water for a day or two. Let it dry out if needed.

Usually plants rot from the bottom up don't they?
I would only cut the soft areas off, if you hit firm branches stop there.

Once its above 50 degrees at night I'd set it outside.