I'd be reluctant to put anything ON TOP of the slab to prevent moisture
from accumulating on top of the slab. That's because whatever you put
on top of the slab WON'T prevent water from rising up into the slab, and
the more water there is in there, the greater the liklihood of
freeze/thaw damage if it gets cold enough in the winter for that slab to
freeze. (I don't know where you live.)
Another possible source of that moisture is condensation. A concrete
slab has a lot of thermal inertia. If it gets cold, it'll stay cold for
a long time before warm weather warms it up again. If the weather turns
hot and humid, you can have enough condensation forming on that cold
slab to make it wet.
You might try duct taping some vapour barrier down over the worst area
of the slab when it's dry, and then checking to see if the moisture
forms above or below the plastic when it's wet there. That will tell
you whether the moisture is accumulating from below or from above.
Could the condensation be forming over the entire surface of the slab,
and then draining to the north end of it by gravity? Is your garage
slab smooth enough to test that hypothesis with a ball bearing? I'm
thinking you could let the ball bearing loose on the garage slab, and
see if it rolls to the north end of the slab.
The fact that the paint has peeled off in the area where the water is
worst certainly suggests that the water is rising up from below, but if
that were the case, one would expect mold to be growing both on and IN
the concrete in that area. If it were condensation, you'd have much
less mold growth because there would be no minerals or nutrients in the
condensation that could provide a food supply to the mold.
Your best bet is to wait until you replace the slab, and put a moisture
barrier down over the compacted limestone before you pour the concrete.
That way you're preventing the moisture from even entering the slab, and
dry concrete is happy concrete.