Here's a few options for you that should help. 
First, for individual troubleshooting and assistance, we have a team dedicated to serving accessibility users. Here's an article about that: https://www.google.com/accessibility/blog/post/announce-disability-support.html
Second, I'd like to review some other possibilities:
• There are two services on your Chromebook that may be speaking. One is ChromeVox which is the screen reader which can be turned off by pressing Ctrl + Alt + Z or by disabling in settings. The other is Select to Speak which can be disabled in Settings. Here's more info on that: https://support.google.com/accessibility/android/answer/7349565?hl=en 
• There was a bug in the past regarding apps downloaded from the Play Store on Chromebooks. The issue was that the Play Store apps used a separate screen reader from the one on the main operating system and it could be stuck on even after toggling the screen reader off with ctrl + alt + z. We resolved that bug in a previous version of Chrome than the one currently shipping on the Stable channel. Here's the bug, please let us know if this is what you're seeing: https://bugs.chromium.org/p/chromium/issues/detail?id=761185
• If neither of the above help, please give us more details on your specific issue so that we may open a bug and investigate. Which apps or services are you using when this happens? Does pressing ctrl + alt + z to turn off ChromeVox have any effect?