Some questions that may help:
What type of power supply are you using? Is it center-positive?
If the voltage regulator is hot, then polarity is reversed or there is a short between GND and VCC. But it sounds like you have resistance between them.
Are the two electrolytic capacitors oriented correctly?
Are you sure you have the 74LS247 and 74HC595 chips in the right position and oriented properly? Soldering on the back looks good.