When getting a new phone through Verizon wireless due to a broken one but not at the time of upgrade do you have to pay the full contract price or is there a discount?

You can call customer service and if you have been a good customer in terms of good payment history, etc., they may give you one year contract pricing. Also, if you deal with an indirect agent and have established yourself as a good customer with them, they may take some off the retail price, and a lot of time you still get the mail in rebate!!