Personally, I would not touch the CDs for cleaning unless there is an audible problem.
I found that for problem disks (with heavy fingerprints, etc.) a cheap ultrasonic cleaner made for things like jewelry and glasses can work wonders, if the instructions are followed and it has an adapter for CDs.
For me. the critical instructions are to use warm water and a drop of detergent. After cleaning ultrasonically for a couple of minutes, rinse under tap water and dry with a microfibre cloth. Compared with vinyl records, CDs are remarkably forgiving.
Just a reminder that the data layer is just under the label, so that is the side one should be careful not to write on, or to scratch
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