Assume we have a conductor that has current flowing and its placed in a magnetic field so that it experiences the Lorentz force, and it gains Kinetic energy due to that force. Would drift velocity change the electron's direction thus the current's total direction changes, therefore, changing the Lorentz force direction applied on the conductor?
Yes, sort of. You need a few clarifications.
The Lorentz force due to a magnetic field does not change the kinetic energy of the charges. It changes the momentum (the direction), but not the energy (or the magnitude of the momentum). And as the charges change direction, so too does the direction of the Lorentz force. In the absence of constraints, like the edge of a sample for example, the charges would move in circles.
Drift velocity is associated with an electric field. The magnetic field would not change the average drift velocity of the charge carriers.