I am having the same exact problem and I am really, really afraid.
The white stingy stuff is probably mucus. This is the stuff that lines your intestines and keeps the lining moist and lubricated.
Small amounts of mucus in stool are normal..ecspecially if your constipated...if you notice it for more than a couple of weeks, I would see a dr.
More fibre in your diet could relieve the constipation.
I highly doubt you have worms but what you describe.
Hope this helps....