Neosporin is fine, but peroxide is irritating. It is good for removing blood, but does not disinfect skin and slows healing.
The important question is why is the dog itching. Fleas are the most common reason. Does he have fleas? Are you using a product that kills fleas? If there are fleas, the first step is getting rid of them. There are several good products on the market that can be used once a month. There are a lot of OTC products that do OK if only a few fleas are around, but are not effective if there is a flea infestation. Tick bites are another bug that causes itching and generally are killed by the same products that kill fleas.
If the itching is causing the sores, they are commonly called hot spots or pyotraumatic dermatitis. These are skin infections caused by the trauma of itching and bacteria. Treatment is to carefully clip the hair away from the spot and clean the area with a cleansing solution. Bactine works OK. Then apply a topical antibiotic with steroid. Vet preparations are called Panalog, Dermavet and other generics. Alternatively, Neosporin and an OTC 1% hydrocortisone cream/ointment help. Sometimes systemic antibiotics and steroids are required.
Itchy skin can also caused by mange mites, fungal infections including ring worm, some bacterial infections, skin tumors, food, inhalant allergies,irritating chemicals, etc..
If OTC and home remedies don't work, a vet exam would help find a specific diagnosis and more effective treatment options.
Drooling can be secondary to peroxide, Neosporin or irritants that could be causing the itching.
Let me know if you have follow up questions.