The "professional" way to salvage a disk is to open the jacket and put the cookie in to a known good jacket.
I use a wide putty knife to open the top of a jacket. Some will pop right open, but others take some work.
It is not without it's flaws, and if anyone has better suggestion I'm all ears, but for 5.25" (doesn't work for 3.5") disks I don't want to rip open I have adopted a cleaning method using Dawn dishwashing liquid and water.
First, I scan labels if needed. If I am very careful I can often avoid getting the labels too wet, but assume they will get some moisture in them.
Then I place four drops of Dawn dishwashing liquid inside the jacket around the hub - one inside the left front, right front, left back, and right back. Then put a little bit of water on each side. Just enough so that the soap becomes slippery and the cookie can turn.
Then I slowly turn the cookie by the hub with my fingers (note that sometimes this can be bit sharp) and go over the surface with either my fingers or a q-tip. Repeat until the entire surface is done.
Then carefully rinse out the disk by running water through in around the hub, again left front, right front, left back, right back, sometimes sloshing it around, until it looks like all of the soap is out of the disk.
Then shake/spin the entire thing to get out some excess water trapped in the liner.
Finally prop open the jacket around the hub with q-tips and place in a cool dry place in front of a fan.
Before putting the disk in to a drive, make sure the cookie spins freely, and re-inspect the surface to see if any trouble spots remain.
When I make a flux dump, after this process I usually find there is a little noise in the signal on the first attempt. So assuming the disk does not rip itself apart, I will take at least two flux dumps and keep whichever one looks best.
Also, when I believe a disk is very suspect - such as if it says "Wabash" on it, I will do a "test fit" using an expendable drive that is just sitting out not connected to anything. I will manually move the heads to the innermost track (where damage won't usually mater much) and manually turn the spindle. Any screeching sounds means it needs more attention.
It is a slow process, but I have found it is usually worth it.