I have cracking, redness, itching, burning around my lips and it is my eczema. If the prescription worked, maybe the doctor could give you more? Do you know what kind of med it was and for what? Either way, I find that Aquaphor (in the little tube) works wonders for the dryness, but it won't cure any infections.
You may have an allergy to something that you are eating.
Do you have a habit of licking your lips a lot? If so, that could be the problem.
Another problem may be a fungal one and you could ask the doctor to take a swab of your lib to find out it in fact you do have a bacterial or fungal problem. Fungal infections do need treating with the appropriate antifungal medication.
Other lip problems can be due to lack of Vitamin Bs in your diet and specifically Vitamin B12. You can buy supplements of this vitamin. Your doctor can also take a blood test to check your Vitamin B12 levels.
As well as keeping your lips well moisturised with the lotions that SassyLassie mentions, you could also try and apply natural set yogurt to your lips before going to bed. If this is a fungal problem, the yogurt will help, until you get the swab tests back from the doctor and get the appropriate treatment.
Best wishes.
I would suggest seeing a dermatologist about this.
In the meantime, try putting Aloe Vera lotion on your lips and even some pure, vitamin E oil, both of these are healers and may well help with the dryness as well.