To be clear, you don’t drink any caffeine right now. No coffee, no tea, no dark sodas or Mountain Dew, nothing. Is that right?
Usually, it should be better to be caffeine free. Being caffeine free helps AVOID the very fatigued feeling in the morning. But, since you do feel fatigued, you could experiment and try a very small amount of caffeine in the morning to feel better. You could be starting a vicious cycle though. You already are drugging to go to sleep, then you want to use a stimulant to wake you up. Think about it. Not that many Americans don’t do it, but it does have it’s downfalls.
If you are waking up feeling groggy for a long time in the morning, and you are completely caffeine free, I think there is an underlying cause not being addressed, like maybe your sleep isn’t as restful as it should be. Or, some other physical thing, but I think you said previously you went to the doctor and they did basic blood work.