Yes, you would lose the email address. Your verizon.net address is linked to your HSI or FiOS account. Email isn't provided for wireless plans.
You could keep it by purchasing Verizon Your Domain if you are looking to save money without losing your address. It's not free, but $19.95/year is cheaper than even the cheapest internet plan.
The best advice though is probably to get an email address that is not provided by your ISP. There are several free providers out there (Gmail, Yahoo, and Hotmail just to name a few), and some very cheap custom domain options. None of those would have to change when you change your service provider, meaning you wouldn't have to worry about situations like this one. 😉