Fire Belly Toads go from brown to green often. I have 6 in one aquarium, and right now 2 are brown. There are LOTS of reasons for them to change colour, and you cant narrow it down to stress. They do not require 24/hr attention. Take a breather bud, you are overwhelmed I am sure. All you can do is your best while you look for new homes for them. Look for a local online classified and post them there, and get in touch with some local pet shops. Smaller ones rather than large chains if you can, you will have a better chance at a smaller one.
I can understand the frustration of having lots of animals. I have 3 cats, 6 fire belly frogs in one aquarium, 2 in another, 4 fire belly newts, a blue crayfish, about 25 various fish, 2 dwarf aquatic frogs, and a wife and kids. The animals are all cleaned and fed by myself. What I try to do, is water changes every weekend, for this I have 2 5 gallon buckets where I keep my water with solutions overnight before the water changes. Bi weekly, I pull the substrate and fake plants out, and give them a quick rinse in treated water, and monthly, I pull the water filters and give them afresh filter. In total I have 5 aquariums all with water and filters in them. Other than that, I think I spend 10 minutes each day feeding my creatures.
Like I said, do your best to care for them until you can re home them. You can not blame your animals for your families stress. Your mom will be ok, and getting rid of your pets will not make it better. If your mom is that sick then there is more going on in her life. Wrap your arms around her, give her a big squeeze, tell her you love her, and that you will get through this together.
Good luck with everything, and I hope your family gets the peace it needs! If I lived nearby I would love to take some pets off of your hands and give them a good home. I am up in the great white north!