Are you peri meno? If so, your hormones will be fluctuating and having a low dose of oestrogen can balance this.
It is very unusual for oestegen to affect sleep negatively - it usually improves sleep. It's progesterone that can sometimes disrupt our sleep.
Some do find that it's coming off the progesterone, when on sequential HRT, that causes the low mood for a couple of days. Progesterone can have a sedative effect for some women and some women seem to feel better taking progesterone all the time , however you can't take a conti HRT until post meno.
Dropping off to sleep is a common problem- I assume you have tried all the recommended strategies for helping this eg getting enough exercise through the day, relaxing before bedtime and sleeping in a dark, cool room?
If you want to have continuous progesterone , then why not have a Mirena fitted and then just add oestrogen as or when needed? This can sometimes help in peri.
DG x