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.
The problem is that you have HC type CMOS logic chips, and in the video LS TTL type chips are used. CMOS chips don't work in a circuit that is deliberately made to utilize the characteristics of LS TTL type chips.
So your inputs float unconnected when the DIP switch is off, and the CMOS chips in your circuit do not work properly with unconnected inputs.
LS TTL chips use bipolar transistors and their internal circuitry is such that an unconnected input pin will tend to float high at logic 1 level, as the input impedance is moderate. So they can, but are not recommended to, be left unconnected.
CMOS chips have extremely high input impedance, and have no internal circuitry to default to a certain state. Thus leaving them unconnected can make the voltage drift to any random level, and a finger has low enough impedance to set the pin state quite easily to logic high or low, depending on if you are touching VCC or GND with another finger.
Basically it means that since your DIP switches either connect an input to VCC or leave it completely disconnected, for each DIP switch, you need a pull-down resistor. Almost any value between the range of 1 kilo-ohm to 1 megaohm will work.
CMOS floating inputs are very receptive to stray electric fields.
You need pulldown resistors on the switched_nodes. Use 100Kohm.
Fresh out of university, I assisted a very senior guy debug a 3_rack cabinet logic system: the central_station telemetry receiver for South Florida Flood Control District, monitoring dozens of remote sites out in swamps that would daily report rain, water levels (in the swamps), salt water intrusion (in canals where heavier salt water would become the underlying layer), winds, etc.
We had lots of intermittent behavior. Finally we realized about 20% of the RCA Corp DIP packages, on Augat wire_wrap boards, CD40xx 12_volt logic,
had no power or Ground connections
The clock speed was rather slow (100,000Hz bit rate, from the long_range radio links in the 400MHz RF band) and CMOS needs zero input current once transient charges have charged the nodes, and at +12 volts the NOISE IMMUNITY was very high, so the
LOGIC was getting POWER thru the ESD DIODES
Do not let the CMOS inputs float. Have either a switch tying the input to +5 or to Ground; OR have a (10Kohm) resistor pulling in one direction, and the switch pulling in the other.
Or are those SPDT switches? Cannot be, because the white board does not allow such connectivity.