Customer service should
never
ask you to give up your password, especially over social media, even if it is via a DM. It's okay to verify other pieces of info (for example via the phone you verify a pin), but asking for a password is just unacceptable. I did verify that this indeed was a real twitter account of Verizon's too as they link it from their support page
https://www.verizonwireless.com/support/