It sounds as if you have two things going on.
You seem to be using too much anti set-off spray. Set-off occurs within the first minute or so of printing, as ink will usually set in that time. Mishandling of loads could cause set off but gently lowering and moving them should avoid any problems. Dont sit on loads
Reduce your set-off spray and print 100 sheets then inspect them for set-off, as set off usually occurs at the time of delivery.
Run 200 and inspect.
Run 300 and inspect.
If no problem, you may try to reduce it more.
Its worth doing once as you will not have to repeat the process.
If you reduce ink tack at all, use a thixotropic tack reducer.
If my memory serves me right there are more "water dispersable" spray powdwers available, this may reduce the number of blanket washes. Its a few years since I got my hands dirty.
If the spray is an electrostatic dispersal type, check that someone has not cleaned it out with a blade. They will dump spray afterwards.
You are probably cleaning part of the sheet with some part of the sheet transfer system; sheet smoother, run-down wheels, two-sheet trip, transfer cylinders etc. This will show up on the blanket and thus the print.
Hope this helps.