Seeking advice for overlapping leases and early move in

[Nick W.]

I am the tenant who just moved into a property yesterday night (July 27th in Maryland). We signed a year lease for July '18 - July '19. We paid for the full month of July and security deposit. We were going to move in about a week into July, but were told by the landlord that she made a mistake. The previous tenant's lease went through July. They moved out just before July 27th. 

We had jobs starting in July so we couldn't delay the move until August. She did provide us an empty rental of hers in another city (further away from work) to live in rent and utility free for one week. So that problem was solved but we had to change lots of plans, live without most of our stuff for the week, and had to move what fit in our cars twice. I am grateful she offered a solution though. 

She said she would use the rent we paid for July to cover August instead. She told us the day we moved in (July 27th) that she wants us to pay for the remaining days of July because we moved in early. 

My question is: (1) Do we need to write and sign a new lease? (2) Should we pay for the last 4-5 days of July? (3) Anything else we ought to take care of?

I am resistant to the idea of paying for the rest of July because (1) She didn't mention this until we moved in (2) She never mentioned prorating our rent for our benefit when we were going to move in after July 1st (3) This is all initially her fault for overlapping leases. 

I am willing to pay if it is what would be most fair, but I don't know what is normal or customary here. Thank you for any advice!