Flashing CEL is usuauly a misfire indication which tracks with your P0308. So the question is are the other Service Messages typical when a misfire occurs, I do not know.
But what I would do is get your battery and charging system tested. Often when the batteries start to fail on vehicle in the last 10-15 years that have so many computers employed on the vehicle, they often have problems communicating with each other and really strange things start to occur. Not unusual for certain modules to actualy shut down and go "off line" when battery/charging Voltage issues occur in order to conserve power to allow the engine to continure to run.
The other thing is there have been fuel pump control modules that soft fail and will cause driveability and misfire issues as well.
If you have a Voltmeter, start with the basics, if you can get to a parts store and back, have the battery and charging system tested.
BTW, what is the build date on the vehicle?
How many miles on the vehicle?
Have you had any prior battery or starting related issues?
Good luck.