Well, that is the problem. I have fired a 94 without the foreend, but off a bench where I wasn't holding the gun normally. As Malamute says, the guns are accurate (Winchester didn't make many bad barrels), but with the foreend binding, any barrel heating distorts the barrel.
I suggest the following: Obtain a couple of new foreend screws in case you ruin one. Remove the band and foreend. Make sure the screw will go through the band easily and without bending. Then drill/file the barrel slot and the hole in the wood until the screw will go through the band and the stock with no binding.
Then file/grind the inside of the band if necessary so it is snug on the barrel but not tight. All this may result in some slight lengthwise movement of the foreend. If that bothers you, you can shim it at the receiver, but only enough to snug it up.
With the proviso that all the other rules about barrel condition apply, the accuracy should improve considerably.
Jim