+1 to what Robbie said about FD adjustment.

The shifter and pulling conditions are not related:

If the bike pulls to one side or the other the likely cause is fork or frame alignment.
The alignment of the rear wheel in the DOs (or extremely poor wheel dishing) can affect this but should be pretty obvious.
Less obvious is damage or bending of the fork or frame:

First, make SURE that the front axle is fully seated. Do the fork legs align with the fork crown? With a long threaded rod, see if the axis of the axle is perpendicular to the steerer tube.

Check frame alignment with the string test - a string looping around the lower headset cup and tied to each of the rear drop-outs should be equidistant to the seat tube on the right and the left.

Beyond these very basic measures, you would likely need some more sophisticated tools to determine if there is twist in the frame at the head tube, or at the bb. There are also specialized tools that can also more accurately check and reset the alignment of the dropouts themselves.