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We obtained ultraviolet and optical spectra for 9 M~dwarfs across a range of rotation periods to determine whether they showed stochastic intrinsic variability distinguishable from flares. The ultraviolet spectra were observed during the Far Ultraviolet M~Dwarf Evolution Survey \emph{Hubble~Space~Telescope} program using the Space Telescope Imaging Spectrograph. The optical observations were taken from the Apache Point Observatory 3.5-meter telescope using the Dual Imaging Spectrograph and from the Gemini South Observatory using the Gemini Multi-Object Spectrograph. We used the optical spectra to measure multiple chromospheric lines: the Balmer series from H$α$ to H$10$ and the \ion{Ca}{2}~H and K lines. We find that after excising flares, these lines vary on the order of $1-20\%$ at minute-cadence over the course of an hour. The absolute amplitude of variability was greater for the faster rotating M~dwarfs in our sample. Among the 5 stars for which measured the weaker Balmer lines, we note a tentative trend that the fractional amplitude of the variability increases for higher order Balmer lines. We measured the integrated flux of multiple ultraviolet emission features formed in the transition region: the \ion{N}{5}, \ion{Si}{4} and \ion{C}{4} resonance line doublets, and the \ion{C}{2} and \ion{He}{2} multiplets. The signal-to-noise (S/N) ratio of the UV data was too low for us to detect non-flare variability at the same scale and time cadence as the optical. We consider multiple mechanisms for the observed stochastic variability and propose both observational and theoretical avenues of investigation to determine the physical causes of intrinsic variability in the chromospheres of M~dwarfs.