One realm of the Silvan Elves that was south of Thranduil's realm in Mirkwood was Lothlórien. As with Thranduil's realm, most of the people were Silvan, but the rulers of Lothlórien (Galadriel and Celeborn) were not.
When the Fellowship reaches Lothlórien, we are told that its people speak the language of the Silvan Elves from east of the mountains.
There was a sound of soft laughter over their heads, and
then another clear voice spoke in an elven-tongue. Frodo
could understand little of what was said, for the speech that
the Silvan folk east of the mountains used among themselves
was unlike that of the West. Legolas looked up and answered
in the same language.
The Lord of the Rings, Book 2, Chapter 6: Lothlórien
According to Appendix F, the language of Lothlórien was actually Sindarin but with a heavy Silvan accent.
In Lórien at this period Sindarin was spoken, though with an
‘accent’, since most of its folk were of Silvan origin. This ‘accent’ and
his own limited acquaintance with Sindarin misled Frodo (as is pointed
out in The Thain’s Book by a commentator of Gondor).
The Lord of the Rings, Appendix F: Of the Elves
Lothlórien appears to be thriving when the Fellowship stays there, and it is able to defend itself from the forces of Dol Guldur during the War of the Ring. Like the other elvish lands, it declines in the Fourth Age and appears to be deserted when Arwen goes there to die after the death of Aragorn.
But Arwen went forth from the House, and the light of her eyes
was quenched, and it seemed to her people that she had become
cold and grey as nightfall in winter that comes without a star. Then
she said farewell to Eldarion, and to her daughters, and to allwhom
she had loved; and she went out from the city of Minas Tirith and
passed away to the land of Lo´ rien, and dwelt there alone under
the fading trees until winter came. Galadriel had passed away and
Celeborn also was gone, and the land was silent.
The Lord of the Rings, Appendix B: The Tale of Aragorn and Arwen
It is never described as a "realm", but we are told that some elves from Lothlórien lived for a while in the southern vale of Anduin. They had left Lothlórien with Amroth and Nimrodel to sail to the West from the Bay of Belfalas. Nimrodel was tragically lost in the White Mountains during that journey (as described in the Lay of Nimrodel), and apparently there were other elves who didn't reach the ships. Legolas referes to this when he meets Prince Imrahil.
At length they came to the Prince Imrahil, and Legolas
looked at him and bowed low; for he saw that here indeed
was one who had elven-blood in his veins. ‘Hail, lord!’ he said.
‘It is long since the people of Nimrodel left the woodlands of
Lo´ rien, and yet still one may see that not all sailed from
Amroth’s haven west over water.’
‘So it is said in the lore of my land,’ said the Prince; ‘yet
never has one of the fair folk been seen there for years beyond
The Lord of the Rings, Book 5, Chapter 9: The Last Debate
Any "realm" that may have existed in this area is clearly not thriving at the time of the War of the Ring.