One illustrative method disclosed herein includes, among other things, forming a sacrificial gate structure above a semiconductor substrate, forming a sidewall spacer adjacent opposite sides of the sacrificial gate structure, removing the sacrificial gate structure and forming a replacement gate structure in its place, at some point after forming the replacement gate structure, performing an etching process to reduce the height of the spacers so as to thereby define recessed spacers having an upper surface that partially defines a spacer recess, and forming a spacer etch block cap on the upper surface of each recessed spacer structure and within the spacer recess.