It does make sense, but speading the word doesn't sound right here, because it isn't really used for expressing of opinions such as being a very caring room-mate, it is usually used to mean telling people about an event.
I would use the phrase She's been telling people instead, which means we can no longer say "of my being". Perhaps it is better to say:
She's been telling people that I am a very caring room-mate.
Even though your original sentence was grammatically correct, I think the ammended version sounds a lot more natural.