Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that Savanah has a deep cough and you are out of her usual medication.
Promethazine dm may not be very helpful for her as although it has a cough suppressant (dextromethorphan) it also has promethazine which is an antihistamine which may cause thickening of respiratory secretions, which could make it harder for her to breathe. Promethazine also has sedative effects, and two drugs causing sedation may not be a good idea for an older dog.
Over the counter cough suppressants you can try are Robitussin DM or Benylin Expectorant. Use only products with Dextromethorphin (cough suppressant) or Dextromethorphin and Guaifenesin (breaks up mucous). Never give those products with Tylenol (acetaminophen) or Ibuprofen or a decongestant as those are toxic for dogs. Give 1 teaspoon per 20 pounds of body weight every 6 hours. The side effect most commonly seen is mild sleepiness because of the dextromethorphan.
Do NOT use these over the counter cough suppressants along with the cough suppressants your veterinarian prescribes (which generally contain hydrocodone), rather they are a different cough suppressant to try if you are out of prescription products.
Use humidifiers to moisten the air that she breathes as that is soothing to her airways. You can also take her into the bathroom with you when you run a hot shower or bath so she can breathe in the steam.
Best of luck with your pup, please let me know if you have any further questions.