Options for Changing Cell Phone Provider

Is your contract with ATT on the phone line or the phone itself? If it's on the phone, then pay off the phone and port your number with Verizon. Will your company give you an allowance for the phone and allow you to just stick with ATT?

I just did the swap from ATT to Verizon about 3 months ago with the same situation.

Went to Verizon store.
They opened me a plan.
Made a call to ATT (I presume a special number) to which they canceled my ATT service and swapped my number to Verizon
Got a new phone with Verizon with 100 dollars off for switching from ATT
My old phone (I bought from ATT) I stilled owed on. I got a "Final" Bill from ATT in mail and paid it.

The swap took a total of 30ish minutes in the store.

Now if you are wanting to use the phone you currently have and just bought with ATT it's a bit different.
From my understanding you must pay the phone off with ATT. Then contact them and have them remove the carrier lock.
I believe they must do this, but only can once the phone is fully paid off. I would ask a Verizon Rep.
ATT reps are going to be shady because you are considering leaving for a competitor.

I thought this was going to be a night mare. However, it seems ATT and Verizon worked together to make it quite easy.