The only adjustment you can try would be to make sure that the clutch pedal isn't partially depressing the clutch. Make sure the return spring is there. With the pedal all the way up, reach up in there and see if you can spin the throwout bearing by hand. If it drags on the throwout arm, you can loosen the nut on the rod that connects the pedal to the clutch until the bearing spins freely when the clutch pedal is up.
Now that I've read your post a couple times, check to make sure you don't have a sheared spirol pin somewhere in the driveshaft. Usually that'll cause the tractor to only move in 1st or not at all and make strange noises.
If that's not the problem...
You'll need what I mentioned in the first post. Feel free to ask me any questions, I have a 582 and have done the clutch in it and several other tractors. I'll be reassembling another clutch sometime in the near future. Send me a PM and I'll give you my phone number if you want to call with questions. I can talk you through it and give you pointers if you need.
I could probably do a clutch in one day without turning pressure plates, but I keep some repaired ones on hand so that doesn't hold me up. It'll cost about $125 for the parts I mentioned. You may also need to make a new driveshaft, which isn't too difficult if you have a drill press. You can buy new pressure plates, but they are expensive. Same for the driveshaft. The steel is cheap, so you can save money by making it yourself.
There is some good info here with regards to the stock clutch:
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You might be able to get resurfaced pressure plates and a new driveshaft from him if necessary. Several forum members have lathes...perhaps you could get one of them to true them up for you. Go to the "tools" section of the webpage in my sig line to see what I use to compress the clutch spring. That's a bit difficult without a hydraulic press or some sort of tool like I made.
I should also mention that the service manual is a great help, as well. It's a good time to get one if you don't have one.