Start by asking you if you had a culture to know that you have a strep throat? Anyway, would not worry too much about a single episode of blood tinged mucus. That happened probably because you strained hard to blow your nose. I seriously don't recommend a steroid
shot, unless you had breathing trouble when you went to the ER. The antibiotic shot you had has to be followed up by oral antibiotic therapy of a week at least. Antiseptic mouth washes and saline nasal irrigation
can be helpful. Avoid cold
food. Plenty of hot soups or tea/coffee is fine.