OK, it wasn't easy, but a few hours this afternoon and I was finally able to figure out what was wrong with the Singer. Everything you all said (and other online sources) kept telling me I had a tension problem. I finally found a video which suggested I "feel" the tension on the thread. Once I did, I realized that even on the tightest setting, it was still quite loose.
So I began looking at the tension dial. It ranged from 0 to 9 with a stop at each end. As I was looking it over, I took off the dial, and realized I could adjust the knob with out the dial beyond the single turn range. I tightened the knob until I had what I felt was good tension, and then I placed the dial back on aligned with the medium setting. And it worked perfectly.
Thanks for all the good info. Even if it didn't solve the problem directly, it did lead me where I needed to go.