To answer your question, yes, in order to get a new phone, you must pay off your current balance on your edge plan.
Once your fees are paid, then you can start a new plan with a new phone. Depending on how much you pay depends on when you started this plan.
For more detailed information here's a link to WhistleOut guide to Verizon Edge plan
If you signed up for Edge prior to May 31 of this year(2015), you can make a final upgrade whenever you’re eligible; that is, once you’ve repaid 75% of your current device’s full price.
Customers who joined the Edge plan between June 1 and October 15 of 2014 can upgrade once 60% of their phone is repaid.
If you signed your Edge agreement before June 1, 2014, you can upgrade from your current device once 50% of the full price is repaid.
Unfortunately, once you’ve made your final Edge upgrade, you are then automatically switched to the new device payment program. After this, you can only upgrade after your 24 monthly payments are made, or if you pay out the full balance of your device early.
If your Edge agreement started after May 31, 2015, you already needed to pay 100% of the device cost to upgrade, so the changes won't affect your Edge plan.