I faced this issue really recently. I can tell you my experience, hope it will be encouraging....
DS went from sleeping blocks of 4 hours (consistently 2 or 3 night wakings) to 4, 5 or 6 night wakings just as he reached 15 weeks. He had always fed to sleep. He would go to sleep okay most of the time, but on the transfer to cot he would wake up often and would not resettle, this meant a lot of re- feeding back to sleep and trying to transfer to cot once again, could take about 15/ 20 mins to do this and I could be doing this 3 or 4 times on the worst nights each and every waking. Feeding to sleep was the only thing that worked. I wasn't getting much sleep!
We co slept for a little while but I never slept well with him in the bed and my DH had to sleep in the spare room, which wasn't great - felt more like DH and I were flat mates than husband and wife! I much prefer DS in the cot at the end of our bed.
I did some reading up and discovered the 4 month sleep regression, which I think was part of the issue DS had. I gritted my teeth and got through that (took 2 weeks, we co slept with DH in the spare room, fed to sleep. Then DS returned to waking 3 times a night). After that, I decided it was time to try to break the association between feeding and sleep, so we used the pick up/ put down method with DH being the picker upper/ putter downer so there was no milk in the room(!). We already had a consistent bedtime routine in place (nothing elaborate, just calm change of nappy, into sleep suit, upstairs to darkened room, no noise, no talking, calming atmosphere, etc.). We also make sure that bedtime is between 6 and 7 pm, any later and DS is overtired and its awful. Also, no napping after 5pm and try to make sure DS has 3 naps throughout the day. Not long ones, just 45 mins (ish) around 10am, 1pm, 4pm. We have the jokey saying here that 'sleep breeds sleep' which is totally counterintuitive but true IMHO.
It's still early days for us with pick up put down, and there has been a little crying, but DH is there cuddling DS, so I'm confident he never feels abandoned and we are teaching him how to settle himself, which is an important lifeskill. I now feed DS, put him in his cot, he then wakes up and cries and DH takes over with pick up put down. It went from 1 hour to fully settle the first night to 15 mins last night - we're on our 5th day of pick up put down. Since we started doing this he has woken once each night, so sleeping from 7pm to around 6am with only 1 waking. Nighttime settling for the last 2 nights has been immediate! He's put down awake, gurgles to himself for a few mins then drifts off, incredible. So I am feeling a lot, lot more rested. It could just be a fluke, but it feels like pick up put down has been a positive experience and unlinking feeding from sleeping was definitely the right thing to do for us.
Hope that's helpful and give you some hope. It does sound like 4 month sleep regression to me (even though it was never that great to start with for you!). Hope you can get some sleep, maybe your DH could take the DCs out for a few hours this weekend both days so you can rest. Agree that it's not possible to just sleep in the time that DC is not feeding - at 4 months babies don't just feed and sleep, they are awake, active and exploring the world for good 2 hour blocks throughout the day and naps are less than 1 hour in my experience (both children did this, maybe starlight had a more sleepy baby?).