Can you pass HSV-1 to other parts of your body?

I currently have a cold sore and in my panic, after checking my mouth, I checked my genital area as well (to ensure I didn’t have any sort of bumps down there). Is it possible to spread the infection from my mouth to my genitals through touching my cold sore and subsequently touching my genitalia?

you can but it’s very rare. Just wash your hands often and don’t touch the cold sore. And don’t give oral until it’s completely healed.