Hello, Welcome to Just Answer and thanks for your question. Apocillin will not affect your immune system in any way. Adult acne (rosacea) is treated with low dose antibiotics and medications to help with inflammation. There are many things that can cause skin to itch, which your dermatologist will have to sort out when you see him. Diphenhydramine (Benadryl), loratadine (Alavert, Claritin), cetirizine (Zyrtec) are some other antihistamines you can try to see if they improve the itching for you any better than what you have been taking.
Please read the link I sent you. That is why I sent it, so you could learn about rosacea. It tells all about the condition. Also, one does not become "immune" to cetirizine. Sinus infections are generally viral rather than bacterial. All people have bacteria on their bodies all the time. Strep is no exception. If it isn't causing pain it is normal.
There is no way to give anything but a "superficial" response to such a question. Just Answer is here to give advice and information about disease processes. Diagnoses are done by examination and tests conducted in a personal visit. If your doctors have not found a cause for your itching so far, you will have to see a dermatologist for a diagnosis. I hope that isn't too "superficial" for you. You also mentioned in your first question that "...acknowledged that i was i adult acne case.." then you state that the two family doctors have excluded it. Such contradictions make any attempt to answer you difficult.
There is no way anyone can give you an answer to what you have online. You will need to see the dermatologist for that. There is no way to tell if it is eczema, which just means skin inflammation.
I will opt out and take your thank you for exactly what it is worth.