I don't have any advice, just sympathy. At my high-water mark, I had three kids under 5, three dogs, and two cats. Yikes! I have never been aware of my own limitations.
Now, through attrition due to illness and old age, I still have the three kids, and I have three (different) dogs, and a husband who has been out of the country since September, and I, too, am completely overwhelmed. Honestly, the dogs are more trouble than the kids. I think they are reacting to the change in routine and are acting out because of it. In recent weeks, one that was housebroken has reverted to going in the house (decided she didn't want to go outside in the rain and has now decided it's nice to go in the house no matter what the weather). And they have destroyed: four razors, two hairbrushes, my daughter's headgear, two bags of her rubberbands, two rugs, homework, countless rolls of toilet paper (we have to keep it in drawers now), multiple toys, all my makeup (one dogs can unzip things and she got into my makeup bag--lipstick all over the bed--yeah, I was real happy about that), my card key for work...I could go on.
In my situation, my hubby is coming home at the end of the month and I think their world will feel more normal then, so rehoming isn't a consideration. But if I were going to single parent forever, I would have to think of something. Don't feel guilty about it--they can go to a home that's a better fit where they will be happier. I would ask for vet references for anyone you will be giving a cat or dog to (who is unknown to you).
I know how hard this is--you're doing great taking care of them by yourself for now and will get past this difficult transition.