I just picked up a Trek 470 with 600 (6207 series) components. The rear shifter isn't shifting correctly. When shifting into a bigger cog, if you push the shift lever to a certain spot, it will release about a mm or two of cable when you let go of the lever. This causes you to have to set the cable tension tighter than it should be to get it to stay in a gear.
in indexed mode, it shifts above the gear you want it, then settles back into the gear when you let go of the lever. In friction mode, if you push the lever just a little bit, it will release back to where it started when you let go of the lever, not shifting a gear at all. If you push it a bit further, it will release that same 1-2mm of cable once you let go of the lever.
Do I have a bad shifter? Am I forgetting something in the adjustments? Or were these old 600 shifters really this bad?
You probably don't have the backing plate (small piece that interfaces with the frame) properly situated on the frame. These pieces are hard to find. And I doubt that you will be able to have the local bike shop order one since yours is an old Shimano 600. Best thing to do is remove it from the frame and re-install it. The plate allows the lever to move without letting the lever go to far forwards/backwards.