Radicals vs atoms having unpaired electrons
Yes, a chlorine atom is very much the same as chlorine radical.

With sodium, the two things are technically similar too, but you'll hardly ever encounter a mention of "sodium radical".

Why is that? Well, I guess there is a difference between Na and Cl. A chlorine radical exists in its own right (if only briefly), and is an important intermediate in many reactions. A sodium radical, on the other hand, is pretty much not a thing. Single sodium atoms just don't wander around alone. You may see reactions involving metallic sodium, or ionic salts of sodium, or maybe (not sure about that) some compounds which can pass for covalent, or (unlikely) even $\rm\bf Na^{\pmb-}$... but that would be all.

So it goes.