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Driven largely by multiple ground-based radial-velocity (RV) surveys and photometric space missions such as Kepler and K2, the discovery of new exoplanets has increased rapidly since the early 2000s. However, due to a target selection bias in favor of main-sequence stars, only a handful of transiting planets have been found orbiting evolved hosts. These planets, most of which are giants, hold important information regarding the formation and evolution of planetary systems. In this thesis, I sought to increase the sample of known giant planets orbiting red-giant stars, focusing on data from NASA's Transiting Exoplanet Survey Satellite (TESS) mission, and to improve their characterization. Specifically, I focused on close-in giant planets orbiting (preferably) oscillating low-luminosity red-giant branch (LLRGB) stars. To improve characterization, I developed a method to model planetary transits and stellar signals simultaneously, implementing Gaussian processes to model stellar granulation and the oscillations envelope in the time domain. My results show that the model enables time-domain asteroseismology, inferring the frequency of maximum oscillation amplitude, $ν_\text{max}$, to within 1\%. The method's implementation is open-source and available to the community. Regarding the planet search, I assembled a pipeline, mostly comprised of third-party open-source software and explored a sample of $\sim$40,000 bright LLRGB stars in the southern hemisphere of TESS's field of view. The search identified four planet candidates, two of which are not currently known planets and orbit red-giant stars. Radial-velocity follow-up observations of both these candidates have tentatively confirmed their planetary nature. Finally, I also confirmed the planetary nature of an additional candidate, not part of the above sample, through RV observations.