How can I get my cousins to pay me for caring for their mother?

Who is suffering Memory loss her or you?

Since she is living with you, where is her SS to help pay u for her care and offset room and board. Do you have POA.

Time to tell your cousins that you can no longer care for their Mom. She is beyond your care. THEY need to place her.

The obvious question to me is why isn't your Aunt paying you?

Is there an assumption help by your family (Aunt, cousins & yourself) that you (as female?) MUST provide the care labour? For free?

Imho if however, you chose to take this on.. you can also un-choose.

It's OK to say it's not working anymore - I need to change this. But assuming the cousins will pay you may lead to bitterness. Start planning for yourself.

Honestly, why would you want to continue? Payment isn't going to make your health better.

Give them 2 weeks notice and then drop their mom off to them. It isn't worth your life to continue.

If you've been doing it for five years for no pay they are not going to start paying now.

Unless you have a contract that spells out the job you do and for what pay you are out of luck.
If she is ever hospitalized you can say that you are not responsible for her and give all of her children's name and number for the hospital to contact.
How did she happen to become your responsibility?
And do you have the authority to make decisions if need be? Are you POA?
Is she cognizant?
If so you could tell her that you can no longer care for her and you will have to contact one or all of her children and make arrangements.
If she is not cognizant and you no longer wish to care for her I would contact an Elder Care Attorney and tell them you can no longer care for her. If the children do not respond she will be made a Ward of the State.

You can't. Unless your cousins are incredibly rich and their mother is a) their acknowledged dependant and b) broke, paying for her care is not their responsibility. How did providing her care come to be yours?

You say you've been caring for her for five years, she's now 70: so what led to a 65 year old lady needing her niece's care in the first place?

Give them notice of the day you will quit. As to paying you back pay, without a contract they don't have to pay you anything, and it seems unlikely that they will. So sorry you are going through this.