I think it's good to listen to your gut feelings about this. I totally understand that it would feel impossible to bring it up with T - I would struggle too. But your thoughts about boundaries and it being unnecessary are important and need to be paid attention to especially if the texts continue. It may be that something is going on in your T's personal life that is influencing how she's feeling and texting you is part of that.

My T texts me out of the blue on rare occasions and its really nice. But it's never an ongoing thing and she doesn't ask for a response, so its never a 'how are you' or anything like that. I think it's very tricky ground when a T ask things of a client as it can make it more about them than the client, however unconsciously.