Where did Radagast dwell?

[Gwathagor]

Oh. *crashes and burns* Well, that works, too.

Where did you find that?

[The Might]

I guess you didn't really expect to get a clear answer Legate, so I'll remain speculative.
I would not go for the late writing choice, since it would seem strange that Gandalf wouldn't mention his presence in the area or would not consult him on the problem of Dol Guldur.
I'd just ignore this late writing that isn't in accordance with any other source and go for the old southern location.

[Galin]

Tolkien defined Rhosgobel in his unfinished index (information courtesy of Wayne Hammond and Christina Scull).

The first element appears related to words for 'red-brown' and a root RUS, noting Maedros for example -- though actually a related root beginning with SR- in this case as we have initial rh. According to Appendix E 'RH represents a voiceless r (usually derived from older initial sr-). Gobel maybe contains -pel 'enclosure, fenced field, garth', noting Pelargir 'Enclosure of royal ships'.

But whether my guesses are right or wrong the meaning above is from Tolkien's index in any case, which is the important part.

[Quempel]

The first working map that J.R.R. Tolkien made of Middle-earth shows Rhosgobel situated on the edge of Mirkwood just south of the Gladden Fields. However, a note appended to the essay on the Istari in Unfinished Tales places Rhosgobel farther north, between the Old Forest Road and the Carrock.

http://www.tuckborough.net/dwellings.html

I found this looking for information. I think that Tolkien wasn't set on where it was and had changed it. That said, I always believed Radagast lived in the woods near by Mirkwood, but did not interact much with Elves or Humans but with animals and plants, so he would not be out and about within the normal paths and Rhosgobel could cover the area between southern edge of mirkwood and going further north between the Old Forest Road ect.

[Aiwendil]

Here's a highly speculative idea: suppose we try to reconcile the various quotations by positing two distinct homes for Radagast?

At one time, Radagast lived at his 'old home', Rhosgobel, which was near the borders of Mirkwood at around Beorn's latitude (the northern of the two locations), thus satisfying:

It was Radagast the Brown, who at one time dwelt at Rhosgobel, near the borders of Mirkwood.

In a very late note on the names of the Istari Radagast is said to be a name deriving from the Men of the Vales of Anduin, "not now clearly interpretable." Rhosgobel, called "the old home of Radagast" in The Fellowship of the Ring II 3, is said to have been "in the forest between the Carrock and the Old Forest Road."

Later (by the time of The Hobbit), Radagast left Rhosgobel for some reason and took up residence at another location further south, thus making this statement true when it was made:

perhaps you have heard of my good cousin Radagast who lives near the Southern borders of Mirkwood?

The only quotation against which this supposition seems to go awry is this:

...and some of these had crossed the Mountains and entered Mirkwood, while others had climbed the pass at the source of the Gladden River, and had come down into Wilderland and over the Gladden Fields and so at length had reached the old home of Radagast at Rhosgobel. Radagast was not there; and they had returned over the high pass that was called the Dimrill Stair.

This suggests that Rhosgobel was at roughly the same latitude as the Gladden Fields. It also suggests that (even though Rhosgobel is his 'old home' and he is not there) Elrond expected Radagast to be at Rhosgobel. However, one could imagine that the scouts passed 'over the Gladden Fields' northward and so came to Rhosgobel (though why they would not cross at the more northern pass is a mystery). One could further suppose that Radagast had not definitively abandoned Rhosgobel in favour of the more southern location; perhaps he spent some time at each (and indeed, that may be why he was not at Rhosgobel when Elrond's people arrived).

I don't think I really believe the above proposal, but I thought it might be interesting to consider.

[Legate of Amon Lanc]

Aiwendil, it's true, as you said yourself, that the version you posted does not make sense - however, what if we reversed it? What if Radagast at first lived near the Southern end of the forest, that was his "old home" - he lived there still by the time Gandalf visited Beorn (as Gandalf says to Beorn). This "old home" was what the scouts visited:

...and some of these had crossed the Mountains and entered Mirkwood, while others had climbed the pass at the source of the Gladden River, and had come down into Wilderland and over the Gladden Fields and so at length had reached the old home of Radagast at Rhosgobel

And Radagast was not there. He moved. Why? Because, as we read elsewhere, Dol Guldur was populated again, "and with power sevenfold". It was no longer safe for Radagast to dwell down there, so he moved further North and built "New Rhosgobel" in the land near the Beornings, who probably were his friends and where he felt safer.

And this:

In a very late note on the names of the Istari Radagast is said to be a name deriving from the Men of the Vales of Anduin, "not now clearly interpretable." Rhosgobel, called "the old home of Radagast" in The Fellowship of the Ring II 3, is said to have been "in the forest between the Carrock and the Old Forest Road."

is just how Christopher explains the note of his father. The note was, as it seems to me from this quote, something like "Rhosgobel was between the Carrock and Old Forest Road". Christopher added the words "called the old home of Radagast". Do you see where I am heading? Christopher was not aware that there were two Rhosgobels and identified the "old home" with Rhosgobel from this note. But the Rhosgobel from the note was Rhosgobel#2.

I'm not expecting this to be accepted as a canonical answer, but we have one proverb in Czech: I don't know if an equivalent is in English, but we say "So that the wolf was fed but the goat remained whole". I think this is the case when the wolf was fed - but the goat remained whole.

[Aiwendil]

Another interesting idea, Legate. The problem of course is that then you need to posit two Rhosgobels, which is not anywhere suggested. Whereas if we suppose that the southern location was not called Rhosgobel, we don't have that problem - but we run into trouble with the LotR quote.

I think the 'real' answer is simply that when he wrote the late note, JRRT had forgotten what he'd written many years earlier. It certainly wouldn't be surprising, nor the only instance.