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I don't know the laws in your state, but you can Google your state and tenant rights and see what comes up.
In my state, tenants have rights and landlords have to abide by them.
I would not be afraid of your landlord. Just find out what your rights are and then act in accordance (whether it be a registered letter or putting another lock on the door . . . ) Not advising you to do either one of those things, but find out what you CAN do - don't be a victim.
I would keep small things in the old place until June 30 and turn the keys in at the last possible moment. You have the place till then so why not? Your landlord is anxious to rent the place, but I would do it just to spite the landlord. You already know your security deposit is going to be returned so why not inconvenience the landlord?
I don't know if you are in the NYC area, but on the People's Court, landlords are constantly being sued for not returning security deposits. Your landlord seems like one of the defendants.