Particles in glass-forming liquids may form domains of locally favorable structures (LFSs) upon supercooling. Whether and how the LFS domains would relate to the slow relaxation of the glass-forming liquids have been issues of interest. In this study, we employ tracers of which structures resemble the LFS domains in Wahnström and Kob-Andersen (KA) glass-forming liquids and investigate the translation-rotation decoupling of the tracers. We find that the tracer structure affects how the translation and the rotation of tracers decouple and that information on the local mobility around the LFS domains may be gleaned from the tracer dynamics. According to the Stokes-Einstein relation and the Debye-Stokes-Einstein relation, the ratio of the translational (DT) and rotational (DR) diffusion coefficients is expected to be a constant over a range of T/η, where η and T denote the medium viscosity and temperature, respectively. In supercooled liquids and glasses, however, DT and DR decouple due to dynamic heterogeneity, thus DT/DR not being constant any more. In Wahnström glass-forming liquids, icosahedron LFS domains are the most long-lived ones and the mobility of neighbor particles around the icosahedron LFS domain is suppressed. We find from our simulations that the icosahedron tracers, similar in size and shape to the icosahedron LFS domains, experience drastic translation-rotation decoupling upon cooling. The local mobility of liquid particles around the icosahedron tracers is also suppressed significantly. On the other hand, tracers of FCC and HCP structures do not show translation-rotation decoupling in the Wahnström liquid. In KA glass-forming liquids, bicapped square antiprism LFS domains are the most long-lived LFS domains but are not correlated significantly with the local mobility. We find from our simulations that DT and DR of bicapped square antiprism tracers, also similar in size and shape to the bicapped square antiprism LFS domains, do not decouple significantly similarly to tracers of other structures, thus reflecting that the local mobility would not be associated strongly with LFS domains in the KA liquid.