Hey East Coast,

  I have same kind of situation but u don;t need to worry( In your situation your lawyer isn't familiar with this kind of situation)..

 You will file H1B petition without new I-20 that means when USCIS open your case they will have question "How's this person gonna stay in USA till OCT 1st",, then they will issue RFE (Request for Evidence), its an evidence showing that you will be staying in valid status untill Oct 1st , in our case its NEW I-20.. Once RFE is issued you can send your New I-20 (by then you must have an I-20). If you go for premium processing they will issue RFE sooner than if you don't. From my understanding , earliest they ask may be Mid may then you can send your I-20 within specified period of time. I am saying this all becoz I am in the same situation and I have consult a lawyer who's handled this kind of case before. So just take in mind that once RFE is issued then you must submit new I-20. In your case you are more than safe man, you have already applied for admission so you will be getting your I-20 soon. Tell your lawyer that you can do all this,, if she doesn't know then  tell her to study this kinda stuff in detail and don't make us panic in no reason.

All the best