I am far from an expert sewer, so everything I say should be taken with a grain of salt.
The fray check should work for satin; are you sure you used enough fray-check and left a wide enough seam allowance? For the seams that are coming undone, it looks like the length of those seams aren't long enough to do a straight seam without fastening off the ends in some way; an easy way to fasten off is to back stitch by putting your sewing machine in reverse for a few stitches at the beginning and end of your seam. Your darts mostly look like they just need to be pressed(seriously, pressing can make all the difference for a sewing project), but it also looks like you may be having a slight tension problem with your sewing machine. Some quick ways to troubleshoot a tension problem are re-threading the machine and changing to a new needle. If you want the shoulder seems to be neater, you can hem after sewing the seam, but you would have to hand sew it's a little bit tricky to get the hem to lay right. I hope at least some of this will be helpful.