I had that problem when I first started free motion quilting & it was often because I was moving the fabric faster than the needle & the thread was stretched too tight. I tried to slow down the fabric movement, but I found it was better to increase the needle speed. I sew much better fast than slow.
Don't be afraid to put your foot down! It takes time to get an even stitch length (unless you have a stitch regulator) but the more you practice, the better you'll get.
Also, I avoid free motion quilting on batiks. Their heavier thread count often causes breakage.