One illustrative method disclosed herein includes, among other things, forming a sacrificial gate structure above a semiconductor substrate, the sacrificial gate structure comprising a sacrificial gate insulation layer and a sacrificial gate electrode material, performing a first gate-cut etching process to thereby form an opening in the sacrificial gate electrode material and forming an internal sidewall spacer in the opening. In this example, the method also includes, after forming the internal sidewall spacer, performing a second gate-cut etching process through the opening, the second gate-cut etching process being adapted to remove the sacrificial gate electrode material, performing an oxidizing anneal process and forming an insulating material in at least the opening.