I know you said the iPhone you have is "unlocked", but what you have to determine or have the Apple store or Verizon check for you, is that the phone you have will work on the Verizon network. Only certain iPhone models will work on the Verizon network, regardless of whether it is "unlocked" or not. Verizon should be able to enter the phone's serial number into their system and determine if it is compatible with their network.
As for a contract, you shouldn't need to have one if you are providing your own equipment.
From the Apple website:
"The unlocked iPhone works only on supported GSM networks, such as AT&T in the U.S. When you travel internationally, you can also use a micro-SIM card from a local GSM carrier. The unlocked iPhone will not work with CDMA carriers such as Verizon Wireless or Sprint."
So since you purchased the unlocked version of the iPhone directly from Apple you should explore your options from AT&T.
T-Mobile USA is rumored to finally get the iPhone to work fully on their network. It is about time since the iPhone and iPad is sold thru (DT) T-Mobile EU already, has been for quite some time.