I brought my bike to my LBS (cuz I needed some other adjustments made anyway) and they said it was a borderline case of whether I needed a new tire. They offered to patch it up for free, and I took them up on that. I just wasn't comfortable with riding long distances on it (I am training for my first century in June). So, I decided to get a new tire. As glad as I am that I can change a flat myself, I'd rather be riding than fixing.

I am keeping the old one for winter riding on the trainer, which the LBS guys said it would be fine for.

Thank you all for the great advice!