I went through something similar. A physical problem was disturbing my sleep, but not in a "I haven't slept all night" way. After a few weeks I was so fatigued and brain-fogged that I was very worried for my mental health and ability to do my job. It took a lot of time to piece together what was happening and start taking measures to get my sleep back in decent shape. This was the right call: I still have the underlying physical issue but now I can both 1) be a functional human 2) address the physical issue with a functioning brain, which would have been impossible when I was impaired by lack of sleep.

I wouldn't go as far as to say "most" depression/anxiety is caused by lack of sleep, but by now I am certain that a huge amount of undiagnosed or untreated health issues are caused by stuff like inflammation states, lack of sleep, digestion not doing its thing, muscles doing something wrong, and often more than one of these.

I hope you feel better!