Hi Macsandy52,
If I understand your question, when your iPhone screen goes to sleep (auto-lock is enabled), the audiobook that you are listening to turns off, is that right? One thing you can try is to set your Auto-Lock to Never to resolve this issue. With this setting, you can control when you want the iPhone to go to sleep and your audiobook will not be interrupted.
Sleep/Wake button - iPhone
http://help.apple.com/iphone/9/#/iph3bd014f9
When you’re not using iPhone, press the Sleep/Wake button to lock iPhone. Locking iPhone puts the display to sleep, saves the battery, and prevents anything from happening if you touch the screen. You can still get phone calls, FaceTime calls, text messages, alarms, and notifications. You can also listen to music and adjust the volume.
On iPhone 6, iPhone 6 Plus, and later, the Sleep/Wake button is on the right side:

On earlier iPhone models, the Sleep/Wake button is on the top edge:

iPhone locks automatically if you don’t touch the screen for a minute or so. To adjust the timing, go to Settings > General > Auto-Lock.
Turn iPhone on. Press and hold the Sleep/Wake button until the Apple logo appears.
Unlock iPhone. Press either the Sleep/Wake or Home button, then drag the slider.
Turn iPhone off. Press and hold the Sleep/Wake button until the slider appears, then drag the slider.
Take care,