Aside from using blue painter's tape for marking, and a walking foot (helps a *lot*!), I heavily starch both backing and top before layering. Starch stabilizes the layers so they are much less likely to stretch and distort as you machine quilt. In my case, at least, heavy starching is imperative if I want to do cross-hatching; without it, I would get tucks and puckers when crossing over another line of stitching (because the fabric would have stretched as I worked with the quilt).
For backing fabric, I like a 1:1 solution of Sta-Flo liquid laundry starch and water, "painted" onto backing with a wide wall painting brush, toss in dryer, then iron with steam. This is quite a heavy starch solution. For the top I like to spray several layers of spray starch.
If your quilt is already layered, you can still add starch with the spray starch method. Lay quilt out on the floor on top of a large flat sheet to catch overspray. Spray from edges towards center to minimize overspray. Let dry (a fan speeds drying). Do this several times on one side, then flip the quilt over and do the same on the other side. This will make straight stitching (and especially cross-hatching) easier to accomplish without distortion. But, I would say a walking foot is almost a necessity too, especially for a large quilt.
How is your quilt supported while you sew? The best method is to suspend it. My best guess about the jump to the right and left as you sew is that it is caused by the weight of the rest of the quilt. When a quilt hangs off the side of a table, for example, it is putting stress on the area that you are sewing. This stress can increase or decrease suddenly as you sew when the weight shifts.
Edit: This is not likely, but some of the old battings had cotton seeds in them that could throw off the needle as you stitch. I don't think any of the battings currently on the market have this problem, but I suppose it's possible you are using an older cotton batting.