I would not tear the strips, as it causes the edges to stretch
Iron your fabric open. Then make new folds down the middle as it sounds like the fabric was not folded on the bolt correctly.
Then carefully fold the fabric in half lengthwise, lining up the edges, fold again carefully. Then you should be able to fit it onto your cutting mat. (2 yds folded to 1 yard, folded to 1/2 yard)
You gotta be carefull to line up the edges before cuttying, you might even want to iron it as you go to make the folds sharp and even.