We all have had this at one time or another. When you do your test sew out, you should be sure to stabilize very well. If the test still has gaps, then do another test and increase the stabilizer. Tear away stabilizers are not as strong as cut away stabilizers. Water soluble stabilizers, with the exception of Badgemaster, are easily torn. I did a blog on Badgemaster because of its stronger qualities.
IMHO, it is possible to understabilize, but very difficult to over stabilize. Read my blog on stabilizers to get some more info on that subject. It is time for me to do a follow up on that subject.
BTW, lots of users have said that they seldom use black on their outlines. They use something that coordinates with the design, like brown around a bird or a dark blue around a a blue bird.
Keep the questions coming, someone will get value from your issue.
Pat, The Avid Embroiderer
I hope you'll visit The Avid Embroiderer my blog for some tips and tricks from someone who enjoys sharing her discoveries. Considering selling your embroidery? Check here first! My spreadsheet is just what you need!