Thanks for the advice guys, I went to my parents' house to try to look at the faucet again (I'm trying to fix it for my parents' so I don't get to work on it daily since I live far from them) and even though their basement is finished they had a few holes cut in the ceiling from when some HVAC guys did some work last year. I was able to see the line running through the faucet, and I got lucky that it had a shut off valve! So for now, the leak is stopped without shutting off water to the whole house. This has bought me some time to actually fix it now.
As some of you may remember, the faucet has been a big pain to fix for me simply because the bolt that holds the faucet head on is rusted on and won't come off so I haven't been able to do the simple repair of just replacing the washer. I've tried using PB blaster, tapping with a hammer, and nothing. I spent a little time drilling into the bolt to see if I could just drill it out, but my HSS drill bits may not be hard enough to make quick work of the job (it was taking very long). So I was thinking about just heating the bolt up with a torch and using some penetrating oil (like Kroil oil or even transmission fluid) to see if I that will loosen the bolt up. Are there any concerns I should have when doing this? Or is it better to just try to drill out the bolt? Or should I just replace the whole faucet simply due the stuck rusted bolt?