Oh dear... It sounds like she's been abused a little bit. I'm guessing she might have had a previous owner who abused her somehow with the leash (maybe pulled too hard, I don't know). Try bringing out treats with the leash to show her it's okay. Maybe get lots of small treats, and walk ahead of her, giving her a treat every few feet to show her that the leash is not a bad thing.
Being off a leash isn't *always* illegal (might vary by state, I don't know), but you'd need to make sure that she wouldn't just run away if you let her loose. I'm pretty sure the leash law applies to places like parks, but I could be wrong.
Hope this helps, and I hope you get to keep your puppy, since it sounds like you love her a lot
*Hugs* and good luck!
~TheDeliciousDish