Too much cable tension would allow shifts into the lowest gear but keep you from shifting into the high (smallest cog) gear. I'm assuming the bike would shift into the lowest gear before you did any work on it so if the limit screw was ok then it's ok now.
Start over.
1. Loosen the cable at the derailleur clamp bolt.
2. Shift the rear shifter to the highest gear setting. Click the release lever (assuming its an STI or Ergo) eight or nine times to be sure it's fully into the last click.
3. Turn the crank to let the chain shift to the smallest cog.
4. Turn the rear derailleur's adjusting barrel fully in (clockwise) then back it out 1 or 1-1/2 turns.
4. Pull the cable snug and reattach it to the derailleur clamp bolt.
5. Attempt to shift one click to the second smallest cog. Adjust the cable tension with the derailleur's adjusting barrel until this 1-2 shift is clean and reliable in both directions. Counterclockwise increases the cable tension, clockwise decreases it.
6. At that point you should be able to shift into all of your cogs. Refine the cable tension if needed but only a small amount should be required. Go 1/4 turn of the adjusting barrel at a time.