Can I ask my apt to charge me their new, lowered rent if I already?
Yes you can ask for the lower rent. Of course they can reject your request. Since you havent signed the lease yet, you can also decide to forfeit the application fee and holding deposit and get somewhere else. I would expect them to accept your offer as they are obviously having trouble finding tenants or they wouldnt be lowering the rent. Check other local rents, they may back up your position.