Verizon email accounts are usually POP accounts. This means that instead of syncing a limited number of recent emails in each folder, everything is stored on the iPad. Your Sent, Trash, Archive, and Drafts folder will not automatically clear themselves.
Or if they do clear themselves (I'm not 100% sure on this subject), it may not be quickly enough for your purposes. Especially if you deal with a lot of attachments.
Tap the MailBox button at the top left of the Mail app. Instead of selecting your Inbox from the top of the account list, go down to where the individual accounts are listed, select it and then your various folders, and start cleaning out your older emails. Especially those with attachments.
The Trash folder should offer and Edit, and then Delete All option at the bottom of the listing. I'm not sure about the Sent folder.
Your iCloud account can also be cleaned up this way, but it is less vital. Mail will keep a limited number of emails locally in the iCloud account, downloading older emails only when requested. Most IMAP accounts will behave that way.
Personally, I don't recommend using Verizon or any other ISP as your primary email account. You never know when you might end up switching providers, and then all your email, and online accounts that use that email, will have to be changed.
It is better to choose an email provider that is tied to no other service; provided it's likely to stick around.