In which direction do I unscrew this utility sink faucet to replace the washers?
Both of those screws loosen counterclockwise. Spray some WD-40 on them and let it soak in for a bit.  There is a washer or "o" ring at the base of the spout or in the base of the valve. hit that big nut with some WD-40 and turn it counterclockwise too loosen. you might have to loosen the supply nuts to get an adjustable wrench in there, a pair of Channellocks would just eat the nut up.