Well, do talk to your parents about this type of overthinking and worry. That could be a sign that you have anxiety and phobias and that can continue until you have a hard time enjoying anything because you are always so worried. Talking to a therapist can possibly help. You don't get std's from razors. And you don't get std's in your armpit. STD's are related to sexual activity. Touching things is really not a risk. Body to body contact is a concern at times. But what you describe would not be a risk for STDs.
I agree with everything GuitarRox said.
STDs aren't transmitted by razors, so you're fine there - they are only transmitted by sexual activity. You also don't get STDs in your arm pits or on legs. Even if they have an STD, STDs die pretty quickly once exposed to air.
The only way you get STDs is if you have some kind of sexual activity.
I also agree that you may have anxiety. If you can't talk to your parents, maybe your older sister or aunt or someone. Good for you for asking questions, though. That's how you find out. :)