I need more information.
What is the 1A LED segment like? (part number)
Is there a resistor on each segment? (value)
What voltage on the LEDs?
You might look at the TIP32G transistor. At 1 amp collector current it will need 30mA base current. The 4.7k resistor will need to be more like 470 or 330 ohms to get the base current right. The voltage loss from collector to emitter is 0.2V. There will be 0.2 watts loss so the heat will not be too bad. The TIP32 is very common.
TIP125 Darlington is also good. Much higher gain so the 4.7k resistor is OK. The 10k resistor is inside the transistor. It will be hotter because it will have more voltage lost in the transistor. ( the collector to emitter voltage will be about 1.5V) So there is less voltage to go to the LED. There will be about 1.5 watts in the transistor so HOT.