Question on leaky bathtub faucet

If it is a mixer tap, the spigot has no replaceable parts, and cannot leak "by itself". Very likely one of the two faucet washers, or the valve seats are worn. Best case is that the seats are fine, and you just need new washers. Best to change them both, before the washers wear through, and metal-to-metal grinding begins to groove the seats.

have everybody use the shower first. take it apart when the hardware store opens, you'll have all day to figure it out before they close.

Some are rather tricky to take apart. IT is not the spigot as at most all it has is the diverter. Since you indicated that it was a single handle unit you will need a rebuild kit from the hardware store and it will likely include instructions and maybe even a tool to help you dissemble the thing. You can try to check out the manufacturer's web site and see if you can find the model number so you can get the right rebuild kit.

Note: The idea of having everyone shower and start the project in the morning is a good idea, likely from someone who has been there and done that. :-)