TL; DR: use the Psychology Today and your insurance company’s list to find people who will take your insurance. Email, call, and fill out web forms until you find someone.
I’m afraid this is going to get long, so I’ll try not to ramble and digress.
I have a great deal of sympathy for you and your living situation. Mine is somewhat similar, with people who can’t keep the house clean, but I also can’t kick out. The mess hurts everyone’s mental health, but there is no way I can fix it on my own, and they won’t help.
From what you describe, it sounds to me like depression. You are going to have to take care of yourself before you can really do anything about your family. That situation is not helping you, but getting yourself into a better mental space is going to be the first step.
Finding a therapist is hard. Not “one you click with” or whatever, but just one who is able to see you and takes your insurance. You almost certainly want a psychiatrist as well as a therapist, which can be even more difficult. In some states some psychologists may be able to prescribe drugs, and chemical and talk therapy are probably both necessary.
So, how to find one. As said by others, the Psychology Today site is a good place to start. Also go to your insurance company’s website and they should have information on finding a provider. Cast a wide net.
You may find many of the therapists all have the same address. That may be a clinic, or it may just be a set of offices they share. Do web searches and map searches on the address or phone number. That seems to be the only way to find out sometimes.
If it’s a clinic, they probably have an intake administrator you can talk to. That person is likely to tell you they aren’t taking your insurance anymore, or aren’t taking new patients, but there is no way to know until you check.
If it is individual therapists you may have to contact each of them, or they may share admin staff.
Repeat this process in rings moving out from where you live. Go as far out as you’re willing to travel. Put your name on wait lists. Hopefully you will find something sooner, but it’s good to have eggs incubating. Ideally you can find a combination of therapists and doctors who can talk to you and get you on medication that helps.
You may have to contact a dozen or more places. Keep notes, as the names will all blur together.
If that is not working out, or just seems too daunting (it is; I in no-way joking believe the insurance companies deliberately make it harder than necessary), contact your primary doctor. Tell them you think you are depressed, and you need some help. Be sure to be realistic about the danger, because that may change how they respond, “I don’t want to hurt myself, but I just sleep all day and hide” or “things are bad, and I might do something.”
My primary care doctor has the ability to refer to temporary mental health services. These will be people who can help while waiting for some of those eggs to hatch. I do not know if yours will. Your PCP might just offer you a script for Wellbutrin, and move onto the next patient.
Anyway, good luck. Long, complicated, and discouraging tasks are exactly the kind of thing people suffering from depression are bad at, so if there is anyone you can trust to do the legwork, get help with the process of finding someone.