A method of producing glass sheets includes the step of fusion drawing a glass ribbon along a draw direction into a viscous zone downstream from a root of a forming wedge. The method further includes the step of drawing the glass ribbon into a setting zone downstream from the viscous zone, wherein the glass ribbon is set from a viscous state to an elastic state. The method further includes the steps of drawing the glass ribbon into an elastic zone downstream from the setting zone and stabilizing a region of the glass ribbon in the elastic zone along a width of the glass ribbon extending transverse with respect to the draw direction. A predetermined pressure differential between a first side and a second side of the glass ribbon is used to create the stabilized region. The method further includes the step of cutting a glass sheet from the glass ribbon, wherein the stabilized region inhibits shape instabilities from propagating upstream through the glass ribbon to the setting zone.