@Oneandaquarter im so sorry, you sound absolutely exhausted.
I do empathise as my DD was an awful sleeper and when you’re in the thick of it it just feels never ending.
YANBU to expect your dh to get up in the mornings if you’ve been up in the night. Some solutions maybe/maybe not -
DD sleeps in the same room as DH in the hope he hears her first.
DH stays up late (when he’s not working the next day) and does the first wake ups (assuming you’re not breastfeeding) and you do the 5am one.
If it’s a regular time she wakes, DH sets an alarm for 5am so he hears her when she wakes.
I don’t suppose you have relatives nearby, my mum once had my DD overnight for me when I felt like I was on the edge (she actually bloody slept through for her too)?
I know the recommendations to go to bed early etc. are good but I for one cannot sleep in the day or too early etc. and once I’m awake then I’m awake so that’s frustrating.
I really hope you manage to find a solution so that you can get some sleep, it’s easy for me to say now as mine is a teenager but it does pass, although I look back at those 3 years of not sleeping through with horror, I don’t know how I survived but I did. Good luck @Oneandaquarter