Get a wire dish rack for the sink, and wash them with a soapy fresh sponge in a circular motion. 20-30 seconds per side is plenty. Rinse well with warm water and set them upright in the rack. After about a dozen, the rack will be full and much of the water will have run off.
Gently dry them with a couple of swipes with a clean cotton towel, and set them on edge, in a clean area with good air circulation to finish drying. Start the next batch. Allow them to air dry for at least an hour or 2 to ensure that the labels are completely dry, then put them back into clean sleeves.
If there is mildew, you might spray a little bleach on them a few minutes before the bath, but make sure that it gets rinsed off completely. Weak bleach on a sponge is also a good way to freshen up musty covers, but be sure that everything is completely dry before you stack them up again.
300 records would take pretty much all day with this technique, but you will get a very good result. I have washed hundreds of records this way with great success. The only drawback is that you will occasionally wash the dye out of a label. As I recall, 1970s red/green Atlantic labels and 1950s-60s maroon Columbia labels are weak.