Transistor overheating in seconds when connected to usb power source

• Check the wiring of the transistor, its pinout is B-C-E which is not too intuitional. E goes to GND
• TIP120 is a bipolar transistor, dropping 0.7V when it is fully on. Assuming your motor draws like 2A, this is 1.4W of heat, which is hot by design
• In order to reduce wasting power, you can pick a MOSFET instead of this BJT. Pick e.g. IRFU2405 or AP20N15GI-HF - any N channel incremental mosfet will do. Less on-resistance means less power waste. In short, if you pick a MOSFET rated for 40A, it will essentially have no losses at 1-2A and will make no heat at all.
• Note that the MOSFET has to be a logic-level mosfet, i.e. a device which can turn fully on for less than 5V. The types I am suggesting are logic level MOSFETs.
• You don't need a series resistor between Arduino and the G of a MOSFET. However it is a very good idea to use a pulldown resistor of 1K..10K between G and GND. This ensures that the MOSFET is always OFF if the arduino is not driving its outputs (for example during reporgramming the Arduino).

Not related to the original problem:

• Note that the DPDT relay connection is fishy. It is not very likely that you can actually drive a relay coil directly from a port output, so you might wish to use a small bipolar transistor or a small FET to drive the relay coil. Even if you have the relay coil working, remember that you have a protecting diode missing. Similarly as you need diode for the motor, you need a similar protecting diode for the relay coil, otherwise when turning off the relay, the energized coil may destroy the Arduino.

For conventional current flow going from (+) to (-)

NPN transistor

1. inject current into base causes small current to flow from base to emitter
2. when the base is accepting current it will be about a diode drop more positive than the emitter
3. with the base current there can be current flowing from collector to emitter which can be larger than the base current
4. collector will normally be setup to be more positive than the emitter

PNP transistor

1. extract current from base causes small current to flow from emitter to base
2. when taking current from the base it will be about a diode drop more negative than the emitter
3. with the base current there can be current flowing from emitter to the collector which can be larger than the base current
4. emitter will normally be setup to be more positive than the collector

Keeping the above in mind, which are the proper biasing and current directions for operation of the transistor, you get to select just what represents the power supply to a given circuit. There are many ways to place circuitry around a transistor that abide by the above "rules" and provide a functional circuit.