it sounds as if you are not using the correct size needle for the thread and fabric combination...if your needle is very thin the hole it goes through will (rub) the thread and cause fraying, also, if the eye of the needle isn't the right one for the thread it will cause fraying. you could switch to a larger needle, see if that helps, or one with a rounder eye. or find a finer thread...i would think changing needles would be easier than new thread. they tend to be a bit cheaper than new thread. also, i found if i'm using a needle threader sometimes it will break wound fibers of the thread and cause it to fray and break. i seem to have this problem alot using 100% cotton thread, takes a while to find the correct needle/thread combo. i love hand piecing with silk thread, it is so smooth, strong, and great to work with. and i use 50 weight. i have applique friends who use 100 weight for hand applique, that to me is too thin for piecing, but the 50 weight is perfect. it is on the spendy side though, so i only use it for special projects.
good luck.