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If the device is less than a year old and not been abused our exposed to moisture it can be replaced under warranty. Otherwise you would have to have insurance on the account to replace it. Contact 800 922 0204 for replacement options.
Seems to me that the phone is defective. Call 18009220204 and ask for tech support. if the phone is under a year old Verizon will ship you a factory certified refurbished phone. If its over a year old you can use your insurance if you have any.
It sounds like there's a bad connection between the cable that connects the screen to the main circuit board of the phone. You will most likely need a new phone. As previously mentioned, you can call tech support (800-922-0204) for troubleshooting or possible replacement under warranty. Or you can contact asurion (888-881-2622) for insurance replacement if you have it. If none of these apply, you may want to look into upgrade options.