The datasheet you linked to is the datasheet of the component TB6612FNG made by Toshiba. You are writing about a TB6612FNG pololu DC motor Driver, which probably is a PCB on which the component TB6612FNG is used.
The first hit I got on "TB6612FNG pololu DC motor Driver" was this site (not sure if OP used this) which states 1 A continuous current, which is less than 1.2A. Moreover, a tiny PCB is is nice and small, but useless if they can't loss their (dissipated) heat.
From linked site:
Real-world power dissipation considerations
The TB6612 motor driver used on the carrier board has a peak current rating of 3 A per channel. The peak ratings are for quick transients (e.g. when a motor is first turned on), and the continuous rating of 1 A is dependent on various conditions, such as the ambient temperature. The actual current you can deliver will depend on how well you can keep the motor driver cool. The carrier’s printed circuit board is designed to draw heat out of the motor driver chip, but performance can be improved by adding a heat sink.
Reason why actual current is lower if not cooled well enough
When the IC becomes hotter, the drain resistance of the p-mosfets become larger, giving a bigger voltage drop, resulting in an increase of the "Driver Ron" Sunnyskyguy is already pointing out.
UPDATE in response to comments
If the current lowers gradually, it is due to the increasing \$R_{DS(ON)}\$ of the p-mosfets.
If the current alternates between a certain value and nothing, the TB6612 is triggering its thermal shut down circuit repeatedly.
This thermal shut down circuit disables the outputs when the IC becomes too hot (175°), until the IC has cooled down (by about 20° = thermal hysteresis). But since the motor is still being driven al the time, it will run hot again and enter the thermal protection again very fast, resulting in a hicup mode.
Try cooling the IC with a fan, e.g. an old computer fan. Even a tiny airflow will make significant difference compared to no airflow at all.