Firstly, make absolutely sure you do not have an output shorted to one of the power rails - Check with a multimeter.
How are you driving the LEDs? Each output can sink or source an absolute maximum of 25mA safely, with a total chip current of 85mA. If you are running several LEDs you will most likely be exceeding that. You’ll need to drive them through a transistor,
Heatsinking would cool it down but don’t expect a long lifetime from the chip.
Take a multimeter and set it on the Amps scale. With the chip out, connect the + battery to the + terminal of the board. Connect the plus cable from the multimeter to the - of the board and the minus cable back to the minus of the battery. This should tell you if you have a short on the board. Other than that, my guess would be the same as AndyGadget that the chip is in backwards. Good luck.