I have never had this problem, despite most of my records being from pre-1967, and pawn shop finds, thrift stores, yard sales that were "god awful dirty." I know some styli/carts pick up surface noise more than others. But....you shouldn't be having this problem with a "clean" record. I suspect it's your cleaning method. Dish soap, such as Dawn, will NOT remove all the dirt. From experience, I know this! I have tried just about every cleaner known, from Windex to bathroom cleaners, car soaps, Shout, rinse aids, carpet cleaners, shower cleaners, upholstery cleaners, Simple Green, alcohols and mixtures, etc., on various records and still got lots of surface noise. I finally found a cleaner that works, and always works. It removes oxidized oil (fingerprints 50+ years old), haze, food, bugs, dirt, cigarette stains, grime without damaging vinyl. (I soaked it for 30 days in it; soaked and let dry; let dry for 2 weeks and rinsed) It's available at Walmart and dollar stores, about $4/gallon, and will clean HUNDREDS of records.
This method takes about 4 to 5 minutes per LP start to finish, both to clean the record and have it dry and ready to play. And yes, be sure to use distilled water for your FINAL rinse, before drying your record.
I am amazed by the urban legends that abound for cleaning records. Microfiber is your friend, not your enemy. It is polyester, so it won't scratch. (Compare that to dragging a DIAMOND across a record groove!) It DOES NOT LEAVE LINT behind, but traps lint, just as it snags your finger prints when you handle the cloth. The fibers are very strong and don't shatter, or break into fragments as cotton or paper does. Just make sure your microfiber is CLEAN! Others say that vinyl and water don't mix.......last I heard, PVC was still used for water pipes...both pressure and drain, and for vinyl records.
As for static, you will get that even from a vacuum dry method on vinyl. You will get it just from removing it from a plastic sleeve, or even paper sleeve. You can get it just from removing the record from the platter mat. But one HINT: STATIC is much less on moist surfaces or in moist air. Yes, I use microfiber cloths (the thick ones) to dry my records after washing, too. After your record is clean and dry, use a carbon fiber brush to reduce the static just before play, if you want.
Air drying your records just isn't smart. That allows dust and airborne dirt to collect on your clean record....all over again. Unless you live in a cleanroom environment. (Who does?) If you don't do your final rinse in distilled water, you will leave small water spots. The minerals in water spots are audible on the vinyl, and very abrasive.
With a thorough cleaning, what you will hear is music....and any damaged grooves. But you will KNOW it's damage, not dirt or contaminants.