Actually, it's opposite for me. Since I take my medicine before bed, coffee helps me wake up and feel energized for the day

I just feel sick because I put dairy in my coffee ☹

Sounds like it may be caffeine sensitivity? I probably wouldn't recommend drinking things with high caffeine content :( If you're big on the taste of coffee you might want to try decaf, and if you're looking for a morning wake up drink any kind of black tea with milk is a good way to get a little caffeine but not get you shaky (Hopefully)

Caffeine can promote anxiety in people already prone to anxiety. I cut caffeine out about a year ago because of it.