Welcome to HEALTHCARE MAGIC..
I have gone through your query and can understand your concerns..
As per your complain it seems that you have a bacterial infection only as the white spots
can be typically seen during bacterial infection leading to formation of pus pockets..
Coughing can occur due to irritation and tingling in throat and it can lead to bout of cough reflex.
You need antibiotics for it and you can continue taking Penicillin, while you can also take Azithromycin
You can also take anti inflammatory painkiller like Ibuprofen
Do warm saline gargles and steam inhalations..
You can take cough suppressant containing Dextromethorphan
Sip on warm water and avoid spicy and chilled foods..
The other part of question is that taking Penicillin or any other antibiotic will not cause drug resistence and it is evident in case if you take very irregular doses of antibiotics..
So you need not to get worried about developing resistance to Penicillin..
Hope this information helps..
Thanks and regards.
Dr.Honey Nandwani Arora.