Controlling the light emitted by individual molecules is instrumental to a number of advanced nanotechnologies ranging from super-resolution bioimaging and molecular sensing to quantum nanophotonics. Molecular emission can be tailored by modifying the local photonic environment, for example, by precisely placing a single molecule inside a plasmonic nanocavity with the help of DNA origami. Here, using this scalable approach, we show that commercial fluorophores may experience giant Purcell factors and Lamb shifts, reaching values on par with those recently reported in scanning tip experiments. Engineering of plasmonic modes enables cavity-mediated fluorescence far detuned from the zero-phonon-line (ZPL)─at detunings that are up to 2 orders of magnitude larger than the fluorescence line width of the bare emitter and reach into the near-infrared. Our results point toward a regime where the emission line width can become dominated by the excited-state lifetime, as required for indistinguishable photon emission, bearing relevance to the development of nanoscale, ultrafast quantum light sources and to the quest toward single-molecule cavity QED. In the future, this approach may also allow the design of efficient quantum emitters at infrared wavelengths, where standard organic sources have a reduced performance.