Switching devices doesn't always need intervention by Verizon.... I've swapped out several devices on my own. What two devices are you trying to swap?
Two basic phones: turn off the one you no longer want to use. Turn on the new device, dial *228 option 1 - it will ask for the number you are calling about - enter the number you want to use on the device. It will then ask for a billing passcode, typically the last four of the account owner's SSN, unless the account owner has changed it. If there is an issue, I've usually been transferred to customer service and they have been able to get the job done.
If the new device is a 4G LTE device, you can go onto the verizonwireless.com website and go to "activate a device" and follow the prompts (you'll need a SIM card, free at a local store or they can mail you one).
Post back if you are still having problems, and let us know where the process breaks down for you; we'll be better able to help with more details.