Bamboo skewers are great. I sew on a vintage Singer and used about 3/8" off the end of a skewer to replace the set sdrew on the small belt wheel. Been sewing with it like that for several weeks now with no problem.
You might be veering because of having your markings too close to the needle. If you mark the 1/4" mark all the way to the front of your machine it will help keep the fabric straight.