I think Tolkien said that Sindarin is spoken in Dol Amroth and the Falas as the birth/first language of the general population throughout the Third Age. Unlike in Gondor where it seems to have been a regional and/or class-related language. In some areas (like Minas Tirith) Sindarin is predominant and in other areas it is the principle language of educated people mainly.

I am a terrible scholar--Appendix F, of Lord of the Rings has some things and David Salo's Gateway to Sindarin (University of Utah Press) has a chapter on use of Sindarin at the end of the Third Age, I think, [incidentally some of the old-school language geeks don't like Salo, not sure entirely why?). Also I seem to recall seeing something in one of the issues of Vinyar Tengwar. I no longer have the Salo book or my copies of Vinyar Tengwar, they were in one of a few boxes which we lost when I loved into my current apartment about five years ago!