To clean old LPs found at the Sally Ann and junk shops, I use weak Ivory dishwasher soap and a soft sponge on the LP, quickly flushed with water, then flushed with distilled water from a spray bottle and dried with a clean pre-washed chamois. Works well!
Get as little water on the center label as possible and blot immediately. All the labels on lps that I have found have a pretty good finish and I haven't got any discoloration or lifting.

Keep the lp in a clean sleeve. Before I play one, I track an artist's soft brush, dampened slightly with distilled water from a spray bottle, across the grooves as the turntable goes around. Only takes about 20 seconds.

My daughter has those eraser-like pads that they sold in record stores long ago, which she uses after lightly spraying the LP with distilled water, but my brush with distilled water does a better job, just as quickly.