I have a Mach III - I lather up with shaving cream.....shave down everywhere....then rinse the shaving cream, apply fresh stuff again, and shave against the grain. By this time it is plenty softened up and you can get a great shave. Once that's done, and if you care, run your fingers over your face and chin/neck in all directions to see if any hair grows in other directions that you may have missed, and shave against that hair...
This method takes a little longer, but you may be able to hold off shaving for one extra day by spending the extra 3-5 minutes now....
_______________
Another thing I've learned is to shave before you shower using the method above. Rarely will you get cut, but if you do, just rinse your face while you're showering every 20-40 seconds. Wash your face last. When you get out of the shower you won't be bleeding at all, and you'll be smooth, smooth, smooth.
_______________
One last thing: Mach III's suck ass for trying to trim straight lines, such as sideburns, because it is too loose. I actually have a Gillette Sensor only to trim my sideburns, and then shave everything else with the Mach III.
Try out those tips, you won't be disappointed.