I have difficulty sleeping at night and difficulty staying awake at work during the day. I'll still feel sleepy even if I went to sleep early.

See a sleep doctor. Your symptoms reflect a common issue, but a good sleep doctor (pulmonologist, neurologist) could help identify potential causes, including improving "sleep hygiene", as well as therapy. A sleep study may be needed to further determine whether other aspects (e.g. sleep apnea) are causing some of these symptoms.