Does it have a bobbin tension adjustment screw?

Yes, although it is probably frozen in place :)

Double check the spool of thread and make sure that the little slot that you hook the thread in is upwards. If it's downwards, the thread can catch in it and cause the spinning you describe.

OMG, I never knew it was supposed to be pointed up. Thanks, I learn something new every day and I've been sewing for almost fifty years. That's just like finally reading a sewing machine manual and finding out that knob on the left of the shaft is a thread cutter. LOLOL.

Don't feel bad, I never learned it until about six years ago either. I spent the better part of a day trying to "fix" my machine and then finally actually saw the thread catch in the slot and realized what was happening.

Annie

I have an old 1950's Singer; don't use it much. But I have called their Customer Service 800 number sometimes for help. They've always been very helpful.

Here is a link that might be useful: http://www.singerco.com/

Thanks, everyone! I believe the problem was the bobbin not being wound properly. It didn't look like it was feeding very smoothly. Once it ran out and I refilled it, I have had no problems.