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Isotope shifts (ISs) in atomic energy levels are sensitive probes of nuclear structure and new physics beyond the Standard Model. We present an analysis of the ISs of the cadmium atom (Cd I) and singly charged cadmium ion (Cd II). ISs of the 229 nm, 326 nm, 361 nm and 480 nm lines of Cd I are measured with a variety of techniques; buffer-gas-cooled beam spectroscopy, capturing atoms in a magneto-optic-trap, and optical pumping. IS constants for the D1 and D2 lines of Cd II are calculated with high accuracy by employing analytical response relativistic coupled-cluster theory in the singles, doubles and triples approximations. Combining the calculations for Cd II with experiments, we infer IS constants for all low-lying transitions in Cd I. We benchmark these constants as calculated via different many-body methods. Our calculations for Cd II enable nuclear charge radii of Cd isotopes to be extracted with unprecedented accuracy. The combination of our precise calculations and measurements shows that King Plots for Cd I can improve the state-of-theart sensitivity to a new heavy boson by up to two orders of magnitude.