My name is Melinda,
I signed a rental lease but before taking possession of apartment was able to work out a deal with current landlord. I have been refunded my full security deposit but only half of the first month rent. Am I entitled to a full refund of the first month rent being as I never took possession of the apartment?

No, and you were very lucky to get what you did. Once you sign the lease, and pay them the money, you are legally bound to the entire lease.
So they could have keep the full first months rent, keep what part of the deposit they are allowed to for the breach of the lease, And if they were not able to rent the place to someone else by the 2nd month, bill you for each month till they rent it out.

If they allowed you to get out of this by only keeping 2 weeks of rent, thank you, and be very very happy.