I took a 250mg of hydrocodone a few nights ago, and for the past few days have been gassy and drowsy. Is that a common side effect?

No. This would be a very uncommon occurrence several days following a single dose of hydrocodone. If it persists you might want to see a doctor.
Narcotic Side Effect. Do you mean 2.5 mg? Hydrocodone, a narcotic, is usually paired with medication of a different class -- such as Ibuprofen or tylenol (acetaminophen). Narcotics can cause side effects such as drowsiness and constipation.