You are 100% incorrect. You DO have to give 30 days notice. When a lease expires without a renewal and you have not vacated, the rental agreement automatically converts to a month to month. If you are in a 30 day state, you must give a full month advance notice (plus any remaining days in an uncompleted month.
They must also follow that rule in making changes to the agreement. If they didn't notify you of the rent increase until May 31st, you don't owe the new rent until July 1st. You shouldn't have paid it but you did.
Now, if you want to be out of there, you only have a couple of days left to notify them that you will be vacating. Your last day to move out will be July 31st. Had you given them notice the same day they hit you with the rent increase, you would legally be done with them on June 30th.
Now, because they charged you higher rent when they were not allowed to by law, it doesn't necessarily mean that you can pay them the old rent for July. But I would, and I would do it this way.
Tomorrow, write a letter giving your notice of vacating tenancy effective July 31st. In it, you state that they violated state law by not giving you 30 days notice of a change in the rent, and that they owe you the difference between what you paid them for June and what you legally owed for June. State that you are withholding the difference from July rent. Also warn them that should they attempt to evict you for 'being short' on July rent, you will answer them in court with evidence that they illegally raised your rent by failing to give advance notice.
Mail a copy of the letter by certified mail with proof of delivery and do that tomorrow, but also, make several copies and personally deliver one to the management company.
When you do move out, make sure you have a witness come with you for the walk-through and record it, to document the condition. This can help with any claims they make for keeping security deposit.