I do have a Singer. check your manual on how to thread and wind bobbin thread. Most Sewing machine do have a tension disk on the top of the machine that is use when treading the bobbin. Adjusting the tension on your machine has no affect on this tension. yes thread does get old and will break. Better to buy new thread than keep having problems with old thread. I use thread from connecting tread and both my Singer and Brothers like it.
you should not have to hold the thread while winding the bobbin. Let the machine do the work. The machine should stop when the bobbin is full enough. Also I do not think that the thread should break when the bobbin is full.
Make sure you are using te correct bobbin size as specified in the manual. Singer has several bobbin sizes and not all work in every machine. The size of the bobbin may not make a difference when winding the bobbin but will make a difference when sewing.