I use screens to give myself a break without (much) guilt. I also use a fair amount of childcare.
I'm going to sound like one of "those" parents, but with my 5yo, what I find helpful during weekends/holidays is setting up some activities in the kitchen before they wake up. Something like lego, some colouring in, jigsaws and playdoh. Probably not that useful for older children though.
I don't direct them to these... DC just gets up, wanders in and finds them. But the novelty usually keeps them busy for an hour or so while I sort the baby and then we turn the TV on until it's time to go out. My child can watch hours of TV if left to it, and on really bad days when the baby won't settle, that's what they do.
I book DC into camps/activity sessions if the guilt and screen time are getting out-of-hand. I find this is where having some spare cash (which it sounds like you don't have) comes in useful. So essentially I "outsource" my child for some of the time so I can be a more relaxed and present parent the rest of the time. It's not ideal but with a non-sleeping high needs baby who screams when put down, it's the best I can manage.
My older one is mostly an easy child. They can be boisterous, cheeky and generally annoying, but on the whole they're quite easygoing and laid-back compared to a lot of children their age I come across. The age gap helps. DC1 is rational enough to understand that you can't really argue with a baby.