Can't help with the ROT, but I can help with next years cabbage worms.
I grow brassicas and have suffered with worms in the past, but now I've started using Activated Compost Tea there is no signs of cabbage worms.
I do see lots of Butterflies around but haven't seen any eggs all summer long.

Use row covers to provide barrier from cabbage butterflies. Best thing that ever happened to organic gardeners--easy, cheap, safe, effective. And reusable for several years.

I don't think your pesticide use had anything to do with the dead leaves. Is the cabbage really rotten or are there just dead leaves inside.

Could it be that you got them wet when it was hot and the sun was shining on them and the leaves sunburnt.

I agree with laceyvail. Row covers work for me to keep the carrot rust fly and onion maggots away. You put it on when you plant seeds or seedlings and leave it on all summer. Moisture goes right through it and they get 90% of the light. They keep the soil a few degrees warmer in spring and a few degrees cooler when it's really hot. You can cut them for the size of your row or raised bed. The only drawback is that until they dry they tend to weigh on the seedlings. A few well placed up side down flower pots or hoops solve that problem

Your cabbage has internal tip-burn caused by inadequate calcium. The cause may be soil moisture related versus inadequate calcium. Drought and a period of water can be influential. Mulch the plants and aim for consistent watering.

I do use row covers and a drip irrigation. Keeping soil moist is important to all vegetables.

Here is a link that might be useful: Internal Tip-Burn

My suspicion is the same as oDan's, inconsistent water. I use Excelsior as a mulch and am very pleased with it. 3/4" black poly tubing is an excellent hoop for such an application, ends cut at an angle and shoved a foot into the ground, like so:

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fabric secured by clips made from same tubing, about 15-20% cut out of the radius lengthwise, like so:

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Dan

Thanks so much for all the answers. organicdan was exactly right. I typed "Internal tipburn" into goole, clicked on images and picture #4 looks as if someone took a picture of my cabbages and posted it to the internet! Thanks so much for the help. I will also try and work in some of the other suggestions about row covers and compost tea. I must admit, when I finally got tired of seeing all the cabbage worms I bought some Captain Jacks Deadbug and boy did it work wonders. But I am new to gardening and I don't know all the options yet. Take care!