Close but no cigar, as Bill told Monica. 
It’s a “barred small capital I” in phonemic transcription – like the schwa a sound heard in unstressed syllables, but pronounced slightly further forward than the schwa. And as you note it’s somewhat elided.
The only time “The Family Circus” was ever funny was once back in the 70s when Dolly explained to Jeffy, “You call a lady ‘Mrs.’ is she’s married, ‘Miss’ is she’s not married, and ‘Ms.’ if it’s a secret”.
Of course, like all things, I think this is somewhat regional or time specific.
Growing up in southeast in the 90’s, I’ve almost never heard anyone use “missuz” or “missus” for a woman. To me it sounds very antiquated and would be used if you were putting on some kind of faux accent of high class.
Men were “mister ____.” Women were “miss ___,” with the “i” barely being pronounced at all.
I don’t find that the final sound of Mrs. is always vocalized (as [z]). It depends on what follows it. It it’s followed by an unvoiced consonant, it is an unvoiced [s]. It is only voiced before a voiced consonant or vowel. When standing alone, it depends on the speaker.
In my experience, 90 percent of the use of “Mrs.” or “Ms.” is by children, and when I was a kid, it was invariably elided – MzJones, M’Smith, and the like. You couldn’t tell just by pronunciation which one was meant.