I think missing shims is most likely the binding problem. I regularly refurbish freewheels from old bikes, and the most common problem on older freewhels with no bearing shells is the wrong no of ball bearings. They have to be free to move.

If you want to move to 6 speed the rear forks need a very slight spread. I use two 8mm threaded bolts, with lots of nuts and large washers. Each one is bolted to the rear forks and joining them is a piece of 1/2" copper plumbing tube, 3" long. I cut another piece of the same tube, slit it lengthways and force it into the tube, which makes it stronger.

By turning nuts so they move towards the tube, the forks are spread apart with equal load on each fork. You need to stretch by twice the extra width you need, as the forks do spring back. Take it slowly and measure after each stretch until you have the width you need.

If this isn't clear, I'll take a picture and upload it.