Bicycle Mechanics - Down shifting problem

Hi all,

I just finished installing thumb shifters,(top mount), on my old skool, 6 x 2 Raleigh. The shifters are Shimano SIS indexed 6 and friction 3. My derailer is a Shimano SIS. Shifting from H to L and vise versa on the front works fine. Shifting up to higher gears on the back has no issues. The problem is in shifting the back into lower gears:

6 to 5 - Works fine
5 to 4 - I have to shift slightly past 4 and after it shifts, let the shifter fall back to the #4 detent
4 to 3 - I have to shift all the way to the #2 detent. I can then move back and forth between #2 & #3 and it stays on gear #3.
3 to 2 - See the last shift
2 to 1 - Skips gear 2 and goes straight to 1.

Could turning the adjustment screws make a difference? If so, do I use the H or L screw and which way should I turn it?

Thanks everyone.

Try upping the tension on the cable a little bit. Turn the barrel by the deralleur counter-clockwise.

The limit screws only affect shifting into the topmost and bottommost gears. If it's doing that, no reason to play with them. Sounds like a cable tension issue. Also, how is the routing for the cables? Smoooth?

I slightly tightened the cable and then adjusted the barrel until everything fell into place. Thanks for the help!