Trazodone and gabapentin are examples of non-addictive prescription drugs that make you drowsy and are often prescribed for sleep. I take small amounts of both. Even without alcohol I've always had trouble falling asleep.
These drugs are non-addictive, not benzos nor z-drugs. In small amounts they don't leave you groggy in the morning. I've been on the z-drug and benzo merry-go-round before, no thanks.
Talk to your doc if the herbal cures and melatonin don't work.