Downtube shifters won't stay in low gear on rear derailleur I've finished a bike restoration today and it's my first time using downtube shifters. They work and will span the full range of gears but won't stay in the big cog on the back for long, the shifter itself gets pulled down by the cable tension, I've tried loosening the cable but I'm just not sure how to set it up. Which gear should the bike be in, what position should the shifter be in and how tight do I pull the cable? I've tried tightening the actual shifter against the frame, that didn't help. It was a little D ring that pulls out, it's pretty stubborn so I grabbed it with pliers and managed to bend it off, ghetto repaired for now. I'm assuming it's cable tension.

The H and L screws are ok, I'll try disassembling it if I can find a diagram for my shifter.
Downtube shifters are usually called "friction shifters" and friction is key when fixing the issue you describe. I own and periodically use an old Cannondale with such system. Every now and then I find myself with the same problem as yours. How to fix it? Easy, just tighten the lever of the shifter. 

Mine came with an integrated metal triangle handle (couldn't find its exact name, but I've attached a picture as an example.) So they can be tightened using your bare fingers and on the fly, but some other models might need to use a screwdriver. The H and L limits of the derailleurs will have nothing to do with this problem.