old addresses in the mail app

A lot of apps -- the Phone app, an email app, text messaging app, etc -- use your contacts list as the primary reference on who or what services you interact with. Open up your Contacts app and see if those out-of-date addresses are there, and then edit or delete them accordingly.

The Contacts app relies upon a hidden, contacts list database file. The easiest way to work with your contacts list is to just use Contacts app. If you have your Contacts synced with your Google account, you can also edit your contacts listing remotely by logging into your Google account and going to:
https://contacts.google.com
Or the same applies to if you're using an Outlook email account instead, or whatever online service that also has an extensive online webmail service.

Or if you don't want this new email app you've installed to access any and all of the Contacts list on your phone, go into your Settings >> Apps menu, find and open this email app's entry, and go into the Permissions menu and disable it from using Contacts.

 

Well to be fair you also didn't give much information: you didn't say which email app,, which permissions you have granted it, which email service you were using, and you didn't say that you don't store contacts on your device (which, lets face it, is very unusual). All anyone can do is make best guesses based on the information you provide. And if you reply with the wrong tone (e.g. the first sentence here reads as highly passive-aggressive, which I hope was not your intention) you simply put others off attempting to help you.

So my best guess, given this new information, is that they came from the email account you connected the app to. Android apps are sandboxed so one app can't just read another app's private data (unless you grant it admin privileges, which unless you are rooted you cannot do). Where data are intended to be shared between apps they are stored in particular databases with an API to request access (e.g. contacts, SMS). So if we can rule out the contacts database, and assuming you don't have a copy of your old contacts lying around in the generally-accessible part of the storage (e.g. a vcard file on an SD card, which any app could read), the most obvious source of information is the email account that you connect the app to. That may have a contacts list of its own in the provider's servers, even if you don't sync that with the Contacts app, or conceivably the email app may have gone through that account's mailboxes and compiled its own contacts list from the addresses it found there. Remember that many email apps that offer "extra" services do so by routing your mail through the email app provider's servers and analysing the mail in some way (and they don't always spell out clearly to potential users what they do).

If we can rule out the one standard place where contacts are stored, which you seem to be saying is the case, and the same app does the same thing on more than one device, which makes the "ad hoc" answer of there being some other contact source lying around on your device less likely, then it seem to me that the account you connect the app to is the most likely source. But there's not enough information to be certain, this is just thinking through what's possible.