Swhalen

I was unaware that the guitar even had a battery. Do you think that could cause a situation where it works fine for a time, then goes distorted and quiet, and then, after turning everything off and waiting a bit, goes back to working normally for a time again? 

Yes. Piezo preamps draw very little current. So if the battery is on the edge, it can run for a while before losing enough charge to be problematic. Then when unused, internal chemical reactions occur that bring the battery back up somewhat again. Of course eventually the battery will die and never return, but before that happens it will exhibit the behavior you describe.

Adq
Usually it is consider as a best practice to record acoustic guitar with a mic, not with a piezo. Or to use piezo as addition to main mic sound.

The "piezo sound" usually sucks compared to a nice condenser mic, but piezos don't always have to sound bad. This article explains why there's a problem, and some potential solutions. 
 
This article goes into specific EQ settings that work well with piezo pickups. As luck would have it, all the examples use the ProChannel QuadCurve
 
Also, last February the Braintree release included the "Acoustic Piezo" CA-X Amp Sim. Give it a try...as the description in the Rolling Updates section of the site says, "This amp compensates for the deficiencies of piezo pickups to bring out the true beauty of your acoustic guitar. While not intended for acoustic guitars with built-in EQ, if you can disable the EQ you may find Acoustic Piezo a better alternative. Acoustic Piezo can also be very useful for clean, defined electric guitar sounds, especially with neck humbucker pickups."
 
The same update also included 15 ProChannel EQ presets for acoustic guitar.

Thank you, Craig!

No problem! I'm always glad to contribute what I've learned in the hope that it will benefit others.

Don't forget to unplug your cable from the guitar when you aren't using it as it will drain the battery also. At least that is what I've always been told.

Thanks everybody! I am happy to report that I have swapped out the battery on the guitar and so far it seems as though the problem has been solved. I really appreciate all the suggestions and info.

Kev999
Also try replacing the battery in the guitar.

+1

Kev999
Also try replacing the battery in the guitar.

My Taylor did that to me one time.  Freaked me out too.  Definitely check the battery.  The behavior of a dying one is exactly what you describe.

Also, you should edit the title of this post and add:  [Solved]
 
Considered helpful and polite to let people know when they search they can find an answer to this problem here.
 
Glad you got it cleared up.
Cheers