First off I haven't got one doing that so far. If you suspect the tension, take it apart. It isn't that hard to do. Get good instructions for putting it back together. The main thing with putting them back together is to get the parts going in the right direction and in the right order. I say take it apart because it could be dried oil somewhere in the system. I had one that had so much dried oil on the tiny little pin in the center post that it wouldn't engage the tension any more. The springs can have dried up oil on them and not be as flexible until it is cleaned. I use rubbing alcohol to clean the dis-assembled tensions. On the machine you may risk spots on the finish. Oh and I doubt if you need oil on the tension anyway.
There is also a spring that holds the bobbin holder in place - some times it cruds up with lint or dried oil and doesn't work right. I think I am fighting with one of those on a Singer 401. That machine sews ditches between zz stitches. I know it is not the tension because I traded it out with a different machine.
I would like to know about the lint pin, Tate. What does it look like?