If your pieces are skewing at the ends of the seams, you might need an awl, stiletto or skewer (I have a Purple Thang), to help hold the fabric straight as that last little bit goes under the needle.
One of my machines is hard to control because the 1/4" seam is only hitting the left feed dog causing my fabric to skew. My solution to that problem was to move the needle as far to the right as possible and measure 1/4 inch to the right of the needle and draw myself a line to follow from the throat plate down the front of the machine. You could make the little guide rail of moleskin, painter's tape, sticky notes or whatever instead of drawing a line. I'm a paper piecer also and having anything sticking up gets in the way so drawing a line with permanent marker works better for me than constantly having to put a guide rail up and off again.
I have a little cheapo White sewing machine that runs so rough there is no way to keep an accurate seam. Even paper piecing is nearly impossible unless you go very slow with it. I think the feed dogs go in circles on it--LOL!
Another problem I used to have was sewing long seams accurately. I'd start going faster & faster and not paying attention to the fabric on the bottom so it was slipping farther & farther to the left. The top looked like a good straight seam but I'd completely missed the bottom fabric. Now I only run about 6 inches through at a time and stop to check that my strips are aligned.
If you were taught that it's ok to sew over pins, that could be wrecking your seams also. I know my home ec teacher said it's ok to sew over the pins but she sure doesn't show up to help me dig them out of my bobbin area when my machines decided to eat pins and spit out needles.
Lining yourself up with the needle is a good suggestion. I have also noticed myself drifting to the left which causes my seams to drift.