the knurled nut and dropout outerface have been worn down significantly， and it does not hold the axle in position as firmly as before, should I use a tool to knurble the dropout ? will this get the dropout bent slightly? what tool gonna to use? what else can you suggest?

Your problem seems more related to how you are clamping the axle in the dropout; you should have a toothed washer of some sort underneath the nut, smooth on one side (for the nut to turn against). This will grab the dropout with more authority than the nut alone, while also helping to prevent the wheel walking while you tighten the nut.

Wald used to stamp the good ones out by the thousands, but you may still be able to get some. Model number 318.

Yes, you need a washer under the nut. I would recommend a standard nut (no flange) and a separate knurled washer.

Nut is serrated underneath is it not? you can buy a new nut with the teeth again, , & you can get a spring tempered steel star washer.

Yes, the nut is serrated, but an integrated nut/washer is a known poor solution for a rear wheel, and it's obviously not working for the OP.

Problem Solvers Axle Nuts > Components > Wheel Parts > Hub Small Parts | Jenson USA

Is the better way to fix it assuming you can find one that fits your axle. It has the knurled washer built in.

Seeing as how most of your posts are all about how to kludge together stuff the wrong way due to an apparent dumpster-dive's worth of parts availability, and an unwillingness to purchase new/correct parts.

I'm going to suggest a solution that involves spending $$money$$.

the Surly Hurdy Gurdy device is specifically meant to prevent forward facing horizontal dropout slippage.
It addresses your issue.
see here:
A Side of Hurdy Gurdy with Your Cross-Check | Blog | Surly Bikes

Is the locknut for the axle cone serrated?
You want to "grab" on both sides of the DO.

ANdr0id (post 6) nailed it. You need track huts. Jenson gives them a fancy name, but that is just a track nut; a nut that incorporates a serrated washer. Now, when you get those nuts, there is one more step you have to do. Dribble oil (motor oil, 3-1, doesn't matter) in between the washer and the nut and spin it until you know it has worked its way through. Then put some grease on the threads of your axle. Now, when you apply the wrench you will get some serious clamping force and slippage won't be an issue.

Oh, buy two pairs so when the serrations on the first pair get blunted you can just swap nuts. I have three bikes using those nuts and a bag of spares. Your best bet is to buy them from a bike shop since there are several standards for the threads. Bring your bike in or at least one of the nuts but the bike and its axle is better.

Edit: with good track nuts, the Hurdy Gurdy is hardly needed. If you go tothe track and watch the races, you will see that many racers use nothing to prevent slip except those nuts. They work. And next flat in the rain and the cold (and probably in the dark) you will be cursing that device you have to take off to pull your wheel out.

2nd edit: Bill Kapaun is right about getting new "sharp" locknuts. Again, bring your bike to the shop.

Ben

Coaster brakes almost always have a 3/8" x 24tpi axle... This is a hold-over from American/British manufacturing. Unfortunately, track nuts are difficult to find in that pitch--Porkchop BMX is a good source, as that axle size was common in the early days of BMX.

The integrated (proper term: swivel flange nut, for searching purposes) washer on track nuts are nice, since, as noted above, you can't lose them in the dark.

spring tempering will harden a star washer , there are inside toothed and outside toothed star washers..

auto parts/hardware shop..

Fietsbob, your post has nothing to do with mine. Any type of toothed/serrated separate washer is a good solution. I was speaking of a nut that has an increased diameter, serrated portion on it as a poor solution.