How can I create a SPDT for this motor without using a relay?
The DRV8816 is used quite commonly: -



The power supply range is 8V to 38V and it can supply currents of up to 2.8A (normal recommended levels). There are plenty to choose from and I would recommend this method rather than building from discrete transistors.

See also this for other recommendations and some pointers to devices that may not be suitable. If you are intent on using discrete transistors then google images is your friend. If you go down this route please seek recommendations about a circuit you might choose. This one used MOSFETs and looks OK: -



I'm hinting at MOSFETs because they have much less power loss (and volt drop) compared to BJTs. Darlingtons are particularly hungry on power losses because you cannot properly turn on a darlington to a saturation voltage lower than about 1V. For instance the TIP120 will "drop" about 2V when saturated with 3A flowing according to the Fairchild data sheet. That's potentially a power dissipation of 6 watts per transistor (always 2 used in a H bridge to switch on the motor).

With 3A flowing in an IRF9Z30, the volt drop will be about 0.42 volts and power dissipated about 1.26 watts per transistor. Anyway always read the data sheets for the devices.