Hey, I write a lot of music for songs and lyrics are not my strong point. I have a song close to this one that I call "Mr. Mystery."
I like your style and you are close, but one thing about song lyrics is that is number and sequence of accented syllables should ideally be the same in consecutive or in rhyming lines, for example
Mary had a little lamb, little lamb, little lamb
Mary had a little lamb, her fleece was white as snow
(She) followed her to school one day, school one day, school one day
(She) followed her to school one day, which was against the rules
The second stanza adds the word "she" but otherwise the sequence of accented syllables is clear
1,2,3,4; 1,2,3; 1,2,3; 1,2,3
Mary had a little lamb, little lamb, little lamb
(She) followed her to school one day, school one day, school one day
1,2,3,4; 1,2,3; 1,2 1,2; 1,2
Mary had a little lamb, her fleece was white as snow
(She) followed her to school one day, which was against the rules
So, that is one thing. if you look at songbooks they often write the melody just once but have the lyrics for three verses below it, because all the verses have the same syllabic structure and accents.
Another thing is this - keep your rhyming scheme the same on all verses - A, B, A, B or A, A, B, B for example.
Last, I really like "But my Mr. Right, for right now." But you could also just make it, "Be my Mr. Right - Now."
like you could say ....
It's too soon to start a fight
and I don't care no-how
I don't need a Mr. Right
I just want a Mr. ...
Right Now
I know it changes the meaning a little, but it's a little more on the wild side, too, which can make it more fun.
Anyway - just my 2-cents, I know accented syllables is a hard thing to work out, but it is the one thing that makes songs more memorable.