Greetings Dan,
Thank you for contributing to the Apple Support Communities.
It appears that you used to send email to yourself using Messages on your iPhone, but now they are being received as text messages.
You may be receiving the messages as texts because your Apple ID (email address) is linked to your phone number and iMessage is enabled. This would also cause the speech bubble icon to appear in Contacts.
You can verify this in your iPhone Settings:
You can sign in to your Apple ID on an iPhone by tapping Settings > FaceTime and Settings > Messages > Send & Receive.
On iOS devices, find out which addresses are enabled with FaceTime and iMessage by tapping:
- Settings > Messages > Send & Receive > You can be reached by iMessage at:
- Settings > FaceTime > You can be reached by FaceTime at:
From:
iOS and OS X: Link your phone number and Apple ID for use with FaceTime and iMessage - Apple Support
Check to see if iMessage is switched on, and there are check marks next to your email address in the menus above.
You can un-check your email address if you don't want to use it with Messages.
Note that Messages only supports sending SMS and iMessages, as mentioned here:
While you can't email or print directly from the Messages app, you can do any of these:
Copy and paste the message text into an email and send it.
From:
Send messages with your iPhone, iPad, or iPod touch - Apple Support
As an alternate method, you may consider emailing notes to yourself using the share feature in the Notes app:
In many apps, you can tap Share or 
to choose how to share your information.
All the best,
Jeremy
