yes, it could be strep, go rule it out. Just tell the doctor you only want antibiotics if it is strep. Besides, a doctor can't force you to take medicine, he can only prescribe it. But strep needs treated with antibiotics or it could lead to something worse. And you certainly don't want to pass it on to your baby. Look back in your throat and see if you can see some white spots. That is usually indicative of an infection, but not always so please go to the doctor.
I have taken antibiotics several times while breastfeeding and neither one of us (with either of my kids) ever got thrush or a yeast infection. I take probiotics and eat yogurt and i also give my baby the probiotic baby jarrowdophilus (i get it at vitaminshoppe). you can give that as young as newborns. i've also given florastor for kids if they are eating baby food or juice. You can get that at any pharmacy rite aid, cvs, walgreens, etc but you have to ask for it, they keep it behind the counter although it doesn't require a prescription (my pediatrician recommended it).
if you do test positive, I'd get your baby tested as well.