Thanks for an interesting question, and for contacting me through Just Answer.
You are actually on the right track, but perhaps not in the way you think. The cortisone cream will not kill the bacteria, and thus will not resolve the infection, but cortisones are important in treating ear problems in dogs because of the role of allergies in ear disease. Virtually all ear infections in dogs start with allergies. An allergic reaction to pollen, dust, insect antigens, flea bites, or food antigens causes the ear to become hot and moist. That provides an environment favorable to the growth
of bacteria (and yeast, too). Dermatologists consider any ear infection in a dog to be a symptom of an underlying allergy, and they don't consider their job to be done until they have treated the allergy successfully and reduced the risk of recurrence of the ear infection.
Let's apply this to Delilah now.
First, with the infection recurring so quickly after antibiotics are stopped, I question whether the infection is being fully resolved. It is wise to recheck the cytology before stopping the antibiotics - I want to see a clean microscope slide before I start thinking that I have whipped an ear infection. Oral antibiotics are often necessary to clear ear infections, as well. Some bacteria, particularly rod-shaped ones, are capable of rupturing the ear drum or of penetrating an intact ear drum and infecting the middle ear, where we can't reach very well with topicals.
I typically do cytology on the first presentation, then recheck the dog after two weeks of antibiotics treatment - topical for cocci, topical plus oral for rods. I also use topical steroids for the allergic component, and add oral allergy medications, often steroids, if the dog has either generalized itching or severely inflamed ears. If the infection is not resolved at my two week recheck, by both physical signs (resolution of drainage and discomfort) and cytology, I will either continue for another two weeks (if I am making progress but haven't completely cleared the infection) or change antibiotics (if I am making no headway) and recheck in another two weeks. If it is not resolved at that point, I submit a culture and sensitivity to find out what antibiotic this thing will be susceptible to.
When I resolve the infection, I instruct my client to treat the dog's allergies for life, with topical steroids at one to two day intervals if the ears are the only issue, or adding the client's choice of allergy control if the dog has other issues with their skin.
An OTC cortisone cream will be useful at this stage for the ear flap, but you are unlikely to get it deep enough into the ear to help the ear canal, so a more fluid cortisone preparation is going to be needed. Although it is, strictly speaking, an off-label use, I mix dexamethasone in a propylene glycol base for my clients. This is a trick I learned from a dermatologist in a seminar and it works extremely well when I convince my clients to stick with it.
If your veterinarian is not taking a similar aggressive approach to Delilah's ear problems, then you may be better off seeing another veterinarian. If you feel they are trying hard but are frustrated, perhaps taking some of these ideas in to your veterinarian will help them get on top of the problem for you. If you feel you need to see a specialist, you want to request a referral to a dermatologist - ear problems of this sort are considered to be in the realm of dermatology in veterinary medicine because of their link to allergic dermatitis.
I hope these ideas help. Sorry that it got a little long, but I thought you needed to understand this process and the underlying reasons. If you have any followup questions, please feel free to respond to this post.