I can't stress enough the importance of cleaning heads especially when you are dealing with a quantity of old disks. One degrading disk will deposit material on the heads which can destroy subsequent disks you put in. Floppy disk heads basically make direct contact with the disk surface, so any abrasive material (mold, dust or oxide coating) will scratch or dig a groove in the next disks you use. Of course the problem compounds because the material that it scrapes off only serves to make the heads more abrasive.
I also can't stress enough the importance of making sure the heads are 110% completely dry before trying to read any disks. Even after drying with a dry cloth or swab, I'd say 5 minutes isn't enough, wait at least 30 minutes. In my experience even the smallest little bit of moisture will act like silly putty and pull the disk surface right off - especially with older disks that have started to break down.
I say this from recent experience of trying to archive a large collection of old 360K floppy disks, and before I realized what was actually happening managed to destroy quite a few disks (and possibly a drive too).
In short, inspect every disk before you even put it in the drive. Look for any signs of mold or lines on the surface, and listen closely for any sound high-pitched sounds or scraping and if you hear it pull the disk out immediately! For any questionable disk, I'd highly recommend using a "sacrificial" disk drive, cleaning the heads after each disk and be prepared to destroy the next few disks you put in.
For more reading on the topic, this article expands on some of these points in more detail with photos: