Yes to starch ... especially if you are working with narrow strips (many strips in my avatar finish at 1/4").
Yes to sewing each strip in the opposite order. To make sure I know which order I sewed the previous strip - I leave the leading thread tail on the strip. I always know where I started (those ends are going to be lopped off anyway!).
Some of the problem could also be uneven feed while your sewing (the feed dogs pull the bottom fabric faster than the top fabric feeds - resulting in a slight shift of the top fabric forward). Pinning helps. I use a dual feed walking foot to piece which helps immensely.
Make sure your seams are UNIFORM. Whether uniform 1/4" or uniform scant 1/4" ... as long as it's uniform throughout the whole strip. Eyeball it before you press it. If you see a place where it is off ... rip it and fix it. If you are sewing narrow strips as much as 1/16" will throw the balance off.
Ironing the strips is of key importance!! You have to make sure the seam remains straight when you iron the seams. For me, I can iron a long strip straighter if I iron the seams open as opposed to ironing to one side. I will lay the strip down and fingernail press the seam open first, then press it open first on the inside then again on the outside.
After you sew EVERY STRIP re-measure the piece to make sure that the width of the combined strips is even and straight. If not, rip it and fix it as it will give you grief if you don't. Check for bowing or slanting to one side.
Good luck!