What kind of hoop are you using, for example, I had a Janome 350E and the hoop only connected at the top, so the hoop would slightly move from weight of fabric and just moving. I had this problem trying to embroider on a jacket. So I went to my 200 Janome, the hoop connects sideways and pulls the fabric, perfect every time, never is off, love this little machine. I also use a Bernina and the hoop is connected the same way, in the middle of the hoop with 2 connections.
I almost gave up embroidery because of the issue, if this is not the problem, make sure your stabilizer does not stretch at all. Take a piece of stabilizer and stretch it, if it does stretch, take another piece and turn it on a right angle so nothing stretches. Yes, you might have to use 2 pieces of stabilizer but it is worth it. I always buy a cut away, 2.5 oz or more when purchasing stabilizer. Made my embroidery perfection just like I want.