Personally, I would love for T to reach out like this. For me, it shows care and that I matter. One issue I had with T is that despite knowing I was going through hell (and could have done with a teeny bit of support), T withheld. Which I interpret as not caring etc.

However, I see where you are coming from. It seems to come out of the blue plus, you highlight T's (at times) worrying quirkiness. So, yes, I can see how it would be... destabilising. I know you said you don't want to address it with her but it will be difficult to know what her intention is, unless you do. Maybe she loosened her boundaries so as to show caring and kindness, maybe she sees you are struggling and wants you to know you are not alone and there is someone on your side or yes, maybe it's a temporary... oddity.

As it seems to (understandably) worry you, I'd say check it out with her for piece of mind. You could always initiate with something innocent like how you appreciate her reaching out and that it is helpful etc. and see where the conversation goes (to check where she is coming from in texting you all of a sudden)