Tested positive with strep throat for 3 time in last two months. I have sore throat symptoms but nothing else. Am I a carrier? Am I contagious everyday?

Hard to say. We generally reserve the "chronic carrier" label for those who do not get sick with it, although anyone sick with it carries it for a while. You need to investigate whether you are being reinfected by a third party, your old tooth brush, your dog, etc.Some have resistant strep and need 2 stage therapy. Work the specifics out with your doc.
Treatment. You should be treated with antibiotic for 10 -14 days and after treatment have culture from throat to be sure is negative but if positive it is resistant to antibiotic that you use so antibiotic should be Chang and to be treated and have culture after treatment again.