Sounds like you've developed an allergy or a sensitivity to something or other (note: I am not a doctor). The pubic area is delicate, so you might be sensitive there but the soap and shaving cream are perfectly fine elsewhere.
It could be anything from your shaving cream or soap, to your sister's, or even the comfort strip on the razor blade if there is one.
I doubt that an STD was or even could be passed to you this way, but there's no downside to seeing a doctor. Just tell your parents you need a checkup or, if you really, really can't tell them why, you can claim headaches (I don't love the lying, but if you need to see a doctor and that's the only way it can happen, then sobeit. Better for you to get treated or at least checked out).
In the meantime, Job One. If you are old enough to shave, then you are old enough to have your own razor. Buy one that looks different from your sister's and make sure to tell her, this is mine, please don't use it
. Take it out of the shower when you're not using it if there are any chances of confusion.
Consider a gentler soap or shaving cream or gel. I don't know if Aveeno makes them, but it's worth a Googling. Aveeno makes hypoallergenic products and they are usually fragrance-free (common fragrances can also trigger allergic reactions).