Tightening the little screw on the bobbin casing is how to tighten the bobbin tension. Turn it right to tighten, left to loosen--righty-tighty, lefty-loosey. Just turn it a tiny bit at a time and try it.
You've also discovered why many people always use the same color thread on top & bobbin. Some of your pokies will get sucked up or down into the quilt when you wash it but not all. I don't worry too much about pokies myself but I don't make quilts for show.
If your thread is ravelling and breaking though, I would try a larger needle or a topstitch needle before messing with tensions. The eye is bigger and lets the thread flow through more smoothly. I would also run some lines of sewers aid down the spool or cone you are using. That will lubricate the thread and help it flow through the machine more easily.
I have had batting that caused my thread to shred and I have had fabric--corduroy & a poly/cotton sheet that would shred it if I went very fast. The batting had dirty spots in it and you could hear when it hit an area and just knew the thread wasn't going to make it through. It would hit those spots and bird seed looking stuff would fly up out of the quilt! The poly/cotton sheet seemed like an excellent backing at the time but I think it was too tightly woven (too high thread count).