Does time slow down because the universe is expanding at an accelerating rate?

Yes, time does run slower$^dagger$ for far-away objects, as observed from our point of view, because they recede from us at high speeds. And yes, because expansion accelerates, this time dilation slowly, very slowly, becomes more pronounced. This is a well-known effect, and is always taken into account when doing observations. For instance, when observing distant supernovae, one is often interested in how their luminosities decrease as a function of time. This is called their lightcurve. In order to compare lightcurves at different redshifts, they are usually converted to their restframe, i.e. how they would look if you were "standing next to the supernova".

However, time dilation does not work exactly as you seem to think. At galaxy at a redshift of $z$ has its time dilated by a factor of $1+z$, so time runs twice as slow for a galaxy at, say, $z=3$ than for a galaxy at $z=1$. Galaxies with redshifts larger than $zsim1.5$ recede faster than the speed of light, and time does not at all stop here. Only for $zrightarrowinfty$, i.e. at the beginning of time at Big Bang, does the time dilation approach infinity.

$^dagger$EDIT: I originally wrote "faster" instead of "slower"