Thanks everyone for all the advice.

The flat happened on the same wheel, which is why I thought I might be due for new tires. Of course, I was also feeling a little bit of MomOnBike's thoughts:

Did I just succumb to advertising for some tires that are lightning fast, ride like a marshmallow, will remove 20 lbs from both bike (even a 17 lb. bike) and hips, take sharp corners at 90mph, sparkle like a little girl's princess fantasy, and last for 5 million miles? (change those puppies NOW!)

The tread on the tires looked good, and I ran my hands over the inside. I ended up finding another thorn, and I think (hope) I got most of it out. Otherwise, the tire had no significant holes or tears. I went for a 22-mile ride today on a new tube and everything felt great, so my tube change seems to be working for now!