Is it appropriate to use the TB6621 or DRV8871 motor driver for powering 12V 1 amp motors?
The question you didn't ask: you can figure out the stall current of those motors.  Stall current is roughly \$i_r + (i_r - i_{nl})\frac{\omega_{r}}{\omega_{nl}}\$ where \$i_{nl}\$ is the motor no-load current, \$\omega_{nl}\$ is the motor no-load speed, \$i_r\$ is the motor rated current, and \$\omega_r\$ is the motor rated speed.  So I get around 3.5A to 4A stall current.

Of the two parts you list I like the TI part better, and either part is better than the L298.  I think it'll work for you.  I like it because it has built-in current monitoring, which looks to be programmable, and because the data sheet seems more complete.  The part about the data sheet may just be a consequence of me being more used to TI data sheets, however.

You really don't want to run the motor at stall current if you can avoid it -- so setting the driver current limit to 1A or 1.5A would probably be a Good Idea.