I ended up deleting the account and re-adding it.   Space used by Email went from 2.7Gb to 16Mb.

On my Moto X, running Android 4.4.4, Delete the account from System settings (i.e., not in the Email app) -- under Settings, in the "Accounts" section, find the account with the Email app's icon and name that matches the name in the app.  Tap the account name, then the menu icon, and select "Remove account".   Details may vary on other Android versions.

To add an account to the Email app, click the menu icon, then select Settings", "Add Account" will appear in the right side of the top menu bar.