(a) She put on the blanket / put the blanket on because she felt cold. -- Put the blanket on what? Or do you mean "put on" as if an item of clothing? A blanket is not something one normally "puts on" in that sense, so it may be better to say "She wrapped herself in the blanket" / "She wrapped the blanket around herself".
(b) She covers the blanket because it is cold. -- No.