sign (type) Ms Mermaid at the end of letters but would do a written signature with my first initial
I do exactly the same. When I write a letter on paper, my signature is Initials Surname, but with Mrs Tw2K typed below. When i write an e-mail, I am Mrs Tw2K, because that is what I expect the parents to recognise me best as. When signing planners, I am Initials Surname scribble!
I have never addressed a parent by their first name. I very occasionally do 'X's mother' verbally, as in 'Hello X's mother, is that Mrs / Miss....?', because the surname we have on record may not be that parent's surname today, for a variety of reasons. I would equally not expect a parent to address me by my first name in a school context, though obviously the few I do meet in other contexts call me by my first name in those. I get the occasional 'Mrs Initial of surname'.
Very occasionally, parents or children deduce or discover that I am in fact Dr Tw2K, and address me as that for a while.
In school, we call each other by first names in the absence of children / parents, and as Miss / Mrs whatever in their presence. In a school firmly rooted in its community, it is useful to have a 'now i am in my work persona' signal.