Yes it shifts the chainline - but the derailleur is attached to the dropout, so it moves with it, and the cassette is attached to the hub, which is also attached to the dropout. If the derailleur is shifting properly to the small cog you could detach the triangle completely from the frame and it would still shift properly. No, I did not say you should tackle the hanger. You should focus on what changed, which is primarily the cassette to derailleur relationship.

Also, you need to be more complete and accurate in your description of what you have done. Did you actually detach the cable, and does "adjusting" the limit screw mean you backed it off? Were you able to back it off enough so the derailleur pulleys were directly below the small cog? What happens when you try to shift to the cog? Do the pulleys sit below it (or even outboard) but the chain just makes noise, or does it not move to that position?

Finally, what else if anything beside the cassette did you change? Did you just add spacers to the hub or is it an entirely new wheel? Did you change the chain? Is the large cassette cog larger, requiring a longer chain? Did you adjust the b-tension screw?

We are not there to see it, so you need to make any text description specific enough so we almost feel we are there.