I'm not sure Tolkien was entirely consistent here with the published legendarium. The Silvan language does continue to dominate in Lothlórien. However, by the end of the Third Age Sindarin becomes the dominant language of the Mirkwood Elves with little remaining of Silvan except some place names and perhaps personal names.
I absolutely agree that the Legendarium was not always consistent in regards to Sindarin's evolution through the ages and across the many regions and peoples that spoke it; though I was not specifically speaking to the adoption of languages in my previous post, but rather the broader cultural merging that took place between the two.
That said, in regards to Sindarin in the Silvan realms specifically, Tolkien was never terribly consistent on this issue, even within short periods of time near the end of his life. For example:
"... at the end of the Third Age there were prob. more people (Men) that knew Q. (Quenya), or spoke S. (Sindarin), than there were Elves who did either! Though dwindling, the population of Minas Tirith and its fiefs must have been much greater than that of Lindon, Rivendell, and Lorien.*
*The Silvan Elves of Thranduil's realm did not speak S.(Sindarin) but a related language or dialect"
-The Letters of J.R.R. Tolkien, Letter 347, p. 425
and also:
"... but they still remembered that they were in origin Eldar, members of the Third Clan, and they welcomes those of the Noldor and especially the Sindar who did not pass over the Sea but migrated eastward [i.e. at the beginning of the Second Age]. Under the leadership of these they became again ordered folk and increased in wisdom. Thranduil father of Legolas of the Nine Walkers was Sindarin, and that tongue was used in his house, though not by all his folk."
which then continues to elaborate on Lorien:
"In Lorien, where many of the people were Sindar in origin, or Noldor, survivors from Eregion, Sindarin had become the language of all the people. In what way their Sindarin differed from the forms of Beleriand -- see FR II 6, where Frodo reports that the speech of the Silvan folk that they used among themselves was unlike that of the West -- is not of course now known. It probably differed in little more than what would now be popularly called 'accent': mainly differences of vowel-sounds and intonation sufficient to mislead one who, as Frodo, was not well acquainted with purer Sindarin. There may of course also have been some local words and other features ultimately due to the influence of the former Silvan tongue ..."
but then Tolkien said the following in a recorded discussion from approximately the same period:
"By the end of the Third Age the Silvan tongues had probably ceased to be spoken in the two regions that had importance at the time of the War of the Ring: Lorien and the realm of Thranduil in northern Mirkwood."
-Unfinished Tales, "The History of Galadriel and Celeborn", "Appendix A: The Silvan Elves and their Speech", pp. 256-7
In summary, the differences seem to range from small and dialectal in the case of Lorien to outright Sindarin rejection in the case of some of Thranduil's people, while in other writings Tolkien claimed Silvan had completely disappeared in these realms. I do not have a strong personal opinion (or even preference) on the issue due to so many writings being entirely contradictory, as seen above. In fact, in regards to this exact problem, Christopher went on to say the following:
Nowhere (I believe) is it made clear how the adoption of the Silvan speech by the Sindarin rulers of the Silvan Elves of Mirkwood, as described (in my previous post, as quoted by Otaku), is to be related to the statement cited (above) that by the end of the Third Age Silvan Elvish had ceased to be spoken in Thranduil's realm.
-Unfinished Tales, "The History of Galadriel and Celeborn", "Appendix B: The Sindarin Princes of the Silvan Elves", pp. 259-60
In summary, this is just another instance where the Professor, for all his brilliance, was not always entirely consistent, and it is up to us to research the issue as best we can to develop an informed personal preference as there exists no single right answer. For what it's worth, based on the natural evolution of real languages, I believe it far more likely that both realms spoke a (potentially heavily) accented Sindarin that was understandable by native speakers such as Legolas, but less so by those with lesser proficiency such as Frodo.