Yes, that's what I was understanding that you did. That is a typical method for machine piecing. I have never used that approach for hand piecing. With hand piecing I just start anywhere in the block and add pieces; I don't do the "construct a row, construct another row, then join two rows" thing.
When machine stitching rows together, all pieces are joined because the machine stitches through all of the seam allowances.
Because you are hand stitching, you are not going through all of the seam allowances; you are stopping at the intersections.
The intersections you have circled have 6 pieces meeting each other. Think about a "dot" marking where each piece intersects the other pieces. When I come to this kind of intersection, I make sure my hand quilting thread goes through all of the "dots" on all 6 pieces when I make my backstitch. This is because every time I have come to that intersection, I have secured all the "dots" to each other. The last piece I join, the 6th "dot", will be joined to all the others because by then all the others will have been joined to each other.
What I am thinking is that you may not be doing this. I think the same problem would exist if you were machine stitching two rows together but stopped at every intersection. When you machine sew without stopping, all of the "dots" are caught into one seam. If you were to stop at each intersection, however, you could be securing only some of the "dots" to each other rather than all 6 of the "dots" to each other.
This is really hard to explain, especially since this may not be your problem. It's just that I can see it could be a problem if you are joining a row to a row with hand piecing. Stopping at the intersections is correct for hand piecing. However, I have never hand pieced row-by-row as the website you provided indicates. I piece a block in any old order that seems efficient to me. With your block, I probably would have started with a corner and added on one piece at a time until the block was finished.
The row-by-row method will work. I'm just thinking it needs the adaption of making sure you secure all the "dots" at those intersections where more than 2 pieces meet.
:? Sorry, I just can't think of a better way to explain this at the moment.