Welcome to our community and thank you for your question!
I don't think applying for asylum in another city in France is going to work. They will quickly find out that you already applied earlier on and that you have been registered in Italy before. You probably had the opportunity to appeal against the decision, but the deadline for that has been exceeded most likely long ago.
I don't know what happened that you ended up without papers and without getting deported, but getting in touch with authorities will attract their attention to your case again. I'm not sure if that is good or bad: It's not unlikely that it results in your deportation to Italy, but on the other hand it doesn't seem to be an option to stay in France in a situation like yours.
If you can, get in contact with a counselling center in France that is more familiar with the asylum system over there and ask them for advice. They can tell you more about the options you are having. Basically your stay is "illegal" at the moment and France doesn't feel responsible for your case. So, from my point of view, it's important to have an expert take a **** at the case before you decide on the next steps.