If you are using PLA, you may need to add an additional cooling fan. I've had the nozzle drag issue (but not the shift issue) and it was resolved with addition cooling. PLA loves cooling.
A forum dedicated to the stock LulzBot TAZ 3D Printer
Here's how I did it. I started with a manual solution and then added two 80mm fans attached to the frame and controlled with a DC Motor / Stepper hat on the Raspberry Pi running OctoPrint that controls the printer. The 80mm fan speed matches the speed of the cooling fans on the tool head (controlled by M106 and M107 gcode commands). The hat can control two more fans if I need them.