You need to call the SW back and ask when she can reschedule. If she's not available go ahead and ask to speak to her supervisor. I have found that it is helpful sometimes to get the higher ups involved. They generally have more experience and can make things happen quicker. Be brutally honest too with them. Tell them how depressed you are and how overwhelmed you are. There is no way that you should be taking all this on by yourself. I remember your posts from last year Oesdog, you were a much cheerier, happier, lighthearted person. I can see that you are at the end of your rope. I don't think that anyone could continue on as you have been without finally losing it.
Is it possible for you to look to another county for a placement for Danny? I'd start putting him on every waiting list you possibly can. I would take the advice of colorado chick and advertise for someone interested in nursing care. Again, even if it's to give you the opportunity to relax, sit in you chicken coop and soak up some sunshine, whatever....but you do need to find someone outside the family that you can rely on. Look online for a support group in or near your area and start picking their brains. Perhaps they have someone that they use that would be interested in an extra day of work.
I know how hard it is to find someone in a small town. I live in a small community in the mountains, a little over 1,500 people. I was approached once by a woman looking for someone to provide elder care to a local woman who suffered from alzheimers. Her children lived out of state. I had never done it, wasn't even looking for a job, but my 12 year old son was standing next to me and said, "Mom, you should do it. She needs someone." Ok, I'm a sucker. I told the woman I'd give it a try. Well, long story short, 2 days and 1 night a week I cared for Patsy. She was a pure joy! I cooked, cleaned, helped her shower (mind you, I've never done anything like this before!!). I just did what was needed. I took her to the kids basketball game and brought her to her church, where she had all but been forgotten. I cared for her until her death. I am so glad that I did. My son and I have wonderfully fond memories of our time with Patsy.
Once again, I'm rambling on to let you know that there is someone out there that will respond. Don't stop until you find that person. Don't add anything extra to your plate in the time being. If someone says "Let me know if I can do anything" give them something to do!! Ask them to run to the store for you, or to cook you a meal. Don't be shy. Believe me, nobody wants to trade places with you and they'll feel good knowing that they've helped. Finally, pray! Pray for someone to answer your ad, to be right for the job, and if they don't come tomorrow, pray for endurance!!