SOunds like you get off on your cutting - particularly if your seam allowances are even. Recommendation: If you have many pieces to cut that are the same size, take some blue painter's tape and lay it on the line of your ruler the size you need to cut. Or put the tape along both measurements of your ruler when you need to cut a square (or rectangle) That way you will see immediately if your ruler is not aligned properly on the fabric.
Your square for the HST probably wasn't square either. Word of advice on the method you described: You will have to deal with the bias edges and they distort easily. That's why I prefer to do them 2 at a time from oversized squares.
Don't give up yet. But in the meantime, you could work with panels. No piecing there.