Very interesting. It always struck me as odd that one says "Won't you go?" but one can't say "Will not you go?" In fact one would have to drag it out to "Will you not go?" which sounds downright antique.

It seems to me that the hard and fast rule "always use the possessive" is patently wrong, but in this case the first sentence makes more sense. "Me writing" is just less seemly as a subject, than "my writing," imho, because the evidence is the written work and not the writer.
However, one might say "I needn't prove I am a writer, you can see me writing my new novel." In this way, the action of the writer is the evidence, so the accusative seems better suited to the task.

To say this another way:
"My writing" emphasizes the writing, whereas "me writing" emphasizes the writer.

You cannot desecrate the mundane. There has never been a golden age of English usage before your pure sacred language was muddied in the mouths of those boorish commoners. I am certain that people that care about such tiresome minutia care more for their own self importance than they do for the living language.
I imagine your poor pupils also had to eschew the dreaded dangling preposition. Well it is my language too, and you are fucking it up.This kind of obsequious litigiousness with regard to the Rules of English does more to degrade the language than any misused relative pronoun. A language, who's rules do not change with time, dies.

In the Southern US, I often hear people say "Tuesday week" to indicate that the event is not the coming Tuesday, but the one after.
E.g. :
"I'm leaving for Vicksburg Thursday week, and I need someone to feed my crickets."