Same prob over here, had bought a whole pile of cheaper ones some 15 years ago, they're all f*cked. :/
The better quality ones (3M-Sony) are still good.
Gonna learn me to be cheapass.
But at least over here we still find floppys in the stores.
What also remains OK are the game floppys and the Dos install ones, they seems to be high quality ones. (My original Doom-Falcon3-Xwing-.... flops are still ok!)
Reminds me, must run a scandisk/surface testing on them one of these days, that helps to keep them ok, as everything is rewritten when the surface test is done.
(time time time I need time .....)
Of course I have an image from everyone of them, but it ain't the same as the originals.
[Edit]
Seems as dos6.22 setup will only work from A: ..... means: problem to run from bootable CD.
But you can always use a bootable CD (dos6.22 bootable one made with format on the used bootflop to create the bootfile, so that you can use format c: /s from the CD) and with cd drivers as Scatty sugested, and put a \dos\ backup folder on the CD .....
Or try this stuff:
http://www.ultimatebootcd.com/ and build your custom one.
Buit its seems a lot of hassle for not much.