Motor Driver IC Heats-up
A lot of insight into problems of this sort can be gained from studying the datasheet carefully. A good datasheet will provide clear specifications, examples, etc from which to work from.

In the case of the TB6612FNG, the datasheet is not the best, but gives us the most important details. On the first page these two tables are presented:



We can see there are two similar tables, one for Absolute Maximum Ratings and one for Operating Range. AMR's are there to tell you what you must not exceed under any circumstances, not to use for your design - the normal operating figures are to be consulted for this purpose.  

In the case of the current, even the AMR table gives a figure of 1.2A continuous (notice the other figures are for pulsed operation) and the normal operating figure is just 1A. This means 1.5A is almost guaranteed to cause problems with dissipation - o this note the figure of ~0.9W for a mounted chip should be noted.  

As others have also mentioned, my guess is that there is a problem with your connections/wiring, since the motor does not need anywhere near 1.5A when unloaded. The adapter is dropping under load since it's only rated for 1A, but the problem is not the adapter, rather why the circuit is drawing 1.5A - I would check all connections carefully for shorts/miswiring.

Finally, at the end of the datasheet it says there is a thermal cutout, but it's not guaranteed to save the IC from damage.