Currently up feeding my 5 wo (DC2) and very similar pattern. Once she wakes for 3am feed it takes hours to get her back down. Don’t worry, this is normal, yes it does get better and settle down, and you will sleep again!!
You’re not doing anything wrong. The baby is still adjusting to day/night schedules, breastfeeding is still being established, and they are still very dependent on you and just want to be close to you all the time.
It’s tough on you, I get it, been there!! But you are also adjusting to this new world and the tiredness will even out once your body adjusts too.
I know it’s a horrible cliche and unrealistic but try and make a point of sleeping when the baby does sleep during the day. I didn’t with DC1 as I always felt I had things to get on with once she slept.
With this baby though I’ve made a point of taking a nap at least a couple of times a week when she is sleeping. That way there are a couple of days where I get a catch up and it takes the edge off the sleep deprivation.
Also if you want (need) a break try expressing so that someone else can do a feed for you and again that will be an opportunity to take some rest/sleep.
In a few weeks the baby’s sleep pattern will improve and you will feel it’s more predictable and you can relax once they’re asleep knowing they will sleep until a certain time.
It’s rubbish being up at this ungodly hour though, I agree!!