According to Dr. Gaylon Morgan, Professor and State Extension Cotton Specialist, if the cotton fields dry out very soon we may not be facing as much fiber quality issues as we were thinking.
As long as the sprouted and germinated seed dries down completely, there should be relatively few problems with ginning. There will be some losses due to discoloration/color grade and some loss of seed value depending on the percent sprouted seed. However, fiber characteristics should remain unchanged unless we continue to get rain for a long time.
The picture below is of two ginned samples taken from local fields (no cleaners). You can see the color differences between the two, as well as compared to the normal bright white.
This is the seed from the sample. The dried sprouts have created some extra trash, but the seed will still be valuable: the sprouted seed won't be much of an issue for feed, but will lose some quality for oil.