The sleep or not sleeping well issue has been going on for me for a couple of years and it's awful. I just want to add that I am of the mind that it is lack of progesterone that causes this issue, or more accurately the ratio of progesterone and estrogen being 'out'. I am not on HRT but presume that eventually the HRT will over-ride your own hormones and get the balance right - or at least that's what is supposed to happen I think - but logically, in perimenopause the hormones are up and down and getting balanced, whether with HRT or not might take a while once you add or change something in your life, either medication or otherwise.
It's a complete marathon of a balancing act. Hang in there, things will get better. Keep noticing and learning what's happening for you. I wish there was a quick fix and it might be that things do settle soon for you, if not, keep the faith that they will x