Sleep Apnea, and CPAP are only two things that can influence your sleep. Another one is depression. Perhaps if you are depressed, being relieved of that aspect would give you the rest that you need.

I sleep very well with my CPAP and wake up less often during the night (I have been using it for a couple of months) but I am still sleepy most of the time, and I strongly suspect that for me it is depression, not lack of sleep.

Could you possibly go back to where you did your sleep study and let them know you are still always tired? There can be things besides apnea that interrupt your sleep. They might have some good guidance.