Many athletes new to zone training, report difficulty staying in Zone 2 while running. 

This is totally normal!

Initially, you may have to even walk for portions of your workout to stay in Zone 2. Do not worry, this is not a bad thing!  Even professional athletes will run-walk if their heart rate won't stay under their Zone 2 cap for any number of reasons (fatigue, illness, travel, lack of sleep, more stress than normal, etc.)  

As you consistently do your workouts, you'll find you're able to do more and more running in Zone 2 without having to walk. But you must be consistent and stick to the workouts.

Trust the process; although it may not feel like it, Zone 2 work is where you'll gain the majority of the benefits in your training, and yes, you'll still derive benefits even if you have to do some walking to start. Follow the workouts as prescribed, stay consistent, and you will notice a gradual improvement.