I have a old bike with down tube shifters. I have a chain with the correct size and for some odd reason I can't get the rear derailler to shift into the bottom four (4) rear sprockets on the bike. It will shift them upward toward the spokes but it won't go down toward the bottom four. I adjusted the screws in the back and it just seems that it won't kick it downward. I know this might be a hard one but just curious about how to fix this. I changed the cable, tubing, and I must be missing something.

It's a shimano 105 and the only thing I could think of is the cable? Maybe not the correct one? But it fits and does get tight?

Any help is appreciated