Mushy? Sometimes, yes.
When slum books and love letters were so in way back elementary, I used to write those two words hastily because (1) everybody writes "Take care" or "T.C". before their signatures and (2) I have lived with the notion that saying those words means you are concerned with the recipient's welfare; that you actually care, etcetera, etcetera.
However it dawned on me that a possible "etcetera" is that, saying take care may mean its actual opposite. One can say take care but mean to say "I don't care about you anymore so take care of yourself instead." or "I'm so done with you, stay away and don't ever pester me again."
Pessimistic? Maybe a bit. However, it doesn't erase the possibility that it can mean what I just said.
