Since your device fully power cycles or powers off the only thing I can think of which may help is a factory reset because whatever is happening effects the entire phones operating system.Also, the problem may just be that the 1GB RAM the Moto E has is not enough to run the apps you want.

Go to Setting- security option-tick off the power button instantly locks and disable any screenlock Moto E - Service & Repair Power cycle and check app •Try power cycling the phone•Check the Play Store for any application updates. Try restarting the deviceWhen this occurs, the phone will re-establish connection to network.1. Hold down the power button until you receive a menu.2. Touch Power off3. Press and hold the power key to turn back on4. Attempt to relaunch the browseryou can do hard reset (backup files first)