If you do not get your landlord's permission, you will be considered a holdover tenant, and the landlord can evict you. However, in your case, you will be moved out long before that process gains any traction.
In any case, if you do not have your landlord's permission, and you stay, he can accept you as a month-to-month tenant and hold you liable for the entire month of July's rent.
I would move when you are supposed to. Put your things in storage and stay at a friend's house.
Brendan Ross | Real Estate Attorney and REALTORÂ®
Also, see if your new landlord might have your apartment ready early, but you better start doing it ASAP.
Patrick Jeffrey Realty
It's all been said here, your move date is 6/30.
If your move in date is 7/8 check with you new landlord regarding a rent proration unless you signed a lease effective 7/1.
You MUST speak to your current landlord re an extension to move out later than 6/30, he/she may already have someone signed up to move in on the 1st, which in that case you're looking at storage.
Using a professional Real Estate Agent in the first place would have addressed these issues in a timely way. Call your landlord then a mover! Now!