LEDs only light up when my fingers are near arbitrary parts of the circuit
I see two problems here.
First, you don't have any pull-up resistors, so your switches can't actually provide a high input. 74 series chips frequently have inputs that float high, but I think the CMOS varieties (like 74HC) need a pull-up or pull-down resistor to function properly.
Second, you have a lot of unconnected inputs, which are picking up noise and amplifying it into other parts of the circuit. Hook the unused inputs up to Vcc or ground through a resistor.