At some point your St. Augustine lawn probably will have to be mowed when the grass is wet. According to Zac Reicher, Professor and Turfgrass Specialist at Purdue University, lawn grass can safely and effectively be cut when the thatch and blades are wet from rain or ill-timed irrigation. A few alterations to the process will yield good results.
Mow wet grass with a freshly sharpened cutting blade, as sharp as possible so it will slice cleanly through the grass blade.
Set the blade height as high at it will go; between 3 and 4 inches is best for St. Augustine grass.
Set up the mower so that the clippings are expelled from the side of the mower over the area previously mowed. Disengage or remove any mulching attachment.
Double mow the lawn twice to break up the mounds of wet clippings. Mowing again at the same or slightly lower height will further cut up the clippings and redistribute them over the lawn in finer cuttings, as mulch.