Air carrier flight crew had miscommunication regarding aircraft control during taxi for takeoff.

Narrative

Aircraft X Prior to pushback we briefed that I would be the PF (Pilot Flying). Prior to departing from ZZZ on XXR; before crossing the hold short line; the Captain set the thrust levers and began taxing onto the Runway. While still in motion and with the power still set (somewhere above 1.05 Engine Pressure Ratio) the Captain stated; you have the thrust levers." Given that we were still in motion and in the midst of a Captain initiated tiller turn; I did not move the thrust levers; as I had not been given control of the aircraft. During this turn; the aircraft continued to turn to the left well over centerline and I felt the rudder pedals deflect full right as the Captain was trying to regain/establish centerline control. The plane momentarily straightened then immediately began turning hard right. As we again reached the center line (now heading towards the right side of the Runway) I felt the rudder pedals deflect to the left; shortly after the plane started turning left. As we passed the center line again; moving to the left side of the Runway the Captain set the thrust to idle and brought the aircraft to a stop. The Captain then made a PA asking the passengers and flights attendants to remain seated. Tower asked us to vacate the Runway and upon doing so; we requested an inspection of the landing gear by station ops. Seeing no signs of smoke; fire; damage or deflation; we were able to taxi to the Gate. Tower mentioned that a tire "appeared" flat but inspection determined that it was not. Brake temps were monitored and remained within normal limits. Upon arrival to the Gate; Maintenance was called. At no time during the events of this report was I given nor did I accept control of the aircraft. Photos attached show the condition of the tires after the event."