Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that you and Murphy have had such a tough week.
This does sound like it started with an allergic reaction to his vaccinations, and now he is reacting to a steroid injection given to treat that reaction. Some dogs are more sensitive to steroids than others. For some dogs a small dose is enough to lead to a voracious appetite and drive ti drink water, and then subsequently the need to urinate large amounts very often.
Depending upon the particular steroid he was given this could last 2-3 days (short acting) to as long as 10-14 days (long acting).
Steroids can also make them feel very mellow, and I'm sure all that drinking and urinating is exhausting and may be affecting his electrolyte balance.
It may help to give him some pedialyte to drink to try to rebalance his electrolytes.
DO NOT restrict his water and that could lead to dangerous dehydration.
Rarely all this stress and a high dose of steroids could push a dog with the predisposition to developing diabetes to become diabetic or with borderline kidneys to develop kidney failure.
So if this isn't improving by the 7-10 day time frame it would make sense to check his urine for traces of glucose and blood test him to look for diabetes and kidney failure.
But in most instances he should start to drink less, urinate less, and regain his pep over the next few days.
Please let me know if you have any further questions.