I don't know much about knurling on an automatic lathe, but its really tough going on a manual knurl if you begin with the knurl full width over the part. Its best to begin on a narrow sliver at the outboard end, about .1" wide, go in fairly deep and fast to attempt to get the knurl tracking immediately on the second revolution. Once a good track is established, back off the pressure to a moderate depth position and begin the lateral feed. Ideally, complete the knurl in one pass.
If starting the knurl almost off the end of the part is not possible, the next best solution, IMO, is to tilt the toolholder a little bit, to make the leading edge of the wheel dig in and do most of the work.
303 stainless may want to fleck off due to its free machining qualities. 304 would likely be better. It will harden up a bit too, due to the cold working of the knurl, but 96Dp is pretty fine. If you find the magic depth setting, it should knurl up quick in one pass.