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I think your problem is connecting the 595 outputs directly to the base of a grounded-emitter transistor. If the output tries to go to +5V, the current is whatever the IC can deliver into what looks like a short. Try adding a 470 ohm resistor in series between the IC output and the base. If the IC output goes to +5V, the current will be (+5V - 0.6V)/470 or about 9mA. The 74HC595 spec sheet gives VOH for a 6V supply as very near the supply rail at 7.8mA. If the output gets to 4.8V with a 5V supply, you will get something close to 8mA using 470 ohms. You may need to watch the current gain of the drive transistors with the reduced current.