Bicycle Tutor

When I shift the rear derailler to any gear the chain jumps back and forth between two gears. It doesn't matter which gear I'm in, whichever one I shift to, the chain jumps back and forth between that gear and the next lower one ( or alternately, it may be between the gear chosen and the next higher one). I have tried all of the adjustments but nothing seems to help.
It's a Shimano Deore XT rear derailler. It doesn't have the barrel adjustment at the derailler. I tried adjusting the cable tension by loosening the clamp and using a pair of needle nose pliers to tug on the cable but I'm not sure that did in fact increase the tension.

It sounds like you need a small adjustment on the cable tension which is very tough to get right without a barrel adjuster. Is there an adjuster where the cable goes into the shifter? Sometimes they put one there too. Was the derailleur made without an adjuster or is it missing. It is odd there isn't one there.

It sounds like the XT der is one of the new 'shadow' versions. They don't have a barrel adjuster or a big loop of cable housing where the cable meets the der.
Questions:
Is the chain flip-flopping between the gear you're in and the next highest (smaller) sprocket?
Or is it flip-flopping between the gear you're in and the next lowest (bigger) sprocket?
If you answered yes to the first, that means that the tension of the cable is too slack. Make it tighter.
If you answered yes to the second, it means that the tension is too high. Back it off a little.
Cheers.

By the by, unless it's a really old one, it probably not a freewheel. It's a freehub body that has a cassette on it. Just FYI.