I'm using a 5V 1A DC power supply
That is totally inadequate for the motor. The TI Hbridge is also not a good choice, as it has a Darlington output which will consume up to 4V of the power supply voltage.
Use the A4988 driver, with a 12-24V power supply capable of supplying more than 1 ampere continuously. Be sure to set the current limit on the A4988 driver to 1 ampere/phase or less.