Does the 72 hour law that allows someone to get out of a contract apply to rental agreements?

What you are referring to here is a "3-day right of rescission".  This is a legal tool which provides consumers with the right to cancel certain contracts that they have entered into.  However, this right of rescission is uncommon. It only applies to a select few types of transactions and, unfortunately for you, lease agreements are not among them.  Therefore you are bound by the terms of a lease once it has been signed. Still, see what you can work out with your landlord (management company). It may be willing to agree to an alternative arrangement.

There is no general 72-law giving people 3 days to rescind or back out of a contract. (Consider: if there was, then the *landlord* too could back out within 3 days and choose to not rent to you, even if you'd already planned your move.) It's a common myth that there is 3 days as a general proposition, but there's not; only if the contract itself references some period during which the contract can be rescinded, is that the case. Sometimes, in real estate purchases, there's a recission period; not typically in rentals. Otherwise, a contract is binding once signed.

That doesn't mean you can't try to negotiate, since it's the same management company, and see if they'll let you change locations; however, it is up to them. You don't have a right to get out of the contract.