Field effect transistors (FETs) based on random networks (RNs) of single-wall carbon nanotubes (CNTs) have several technological advantages. However, the low sensitivity (or no sensitivity) of RN-CNT sensors to nonpolar molecules is a problematic, negative feature that limits their applications in the detection of a wide variety of diseases via breath samples. In this paper, we show experimental evidence for the detection of both individual nonpolar molecules and patterns of nonpolar molecules, even in the presence of polar molecules in the same environment. We do so by preparing RN-CNT FETs and functionalizing them with organic films that exhibit distinctive electrical and physical (or mechanical) characteristics. Exposing the functionalized RN-CNTs to representative nonpolar breath biomarkers, and, for comparison, to polar molecules in the gas phase, and monitoring the changes in conductance, work function, and organic film thickness show sensitivity toward nonpolar molecules. We explain this observation by carrier scattering as a result of swelling of the organic film upon exposure to (nonpolar) chemical agents. Hence, the sensitivity towards nonpolar molecules can be tailored, even in the presence of polar molecules, by controlling the scattering of charge carrier through deliberate functionalization of CNTs. As examples for the technological impact of our findings, we describe ways to detect lung cancer and kidney disease using specially designed RN-CNT sensor arrays.