How should I go about telling them how I feel?

Lifes2short, not once did I ever ask my Dad's caregivers to use their own money. Dad had a credit card, so he used that whenever something was needed. For groceries, I used an on-line grocery service that had my Dad's credit card on file.

Wait for it....here is what an actual care recipient had said:
"I don't have the money to pay you right now because I loaned the money to my driver to gamble when he took us to the casino, I will pay you later."

Lol Jeanne, Same answer! We have a consensus!

Feelings are subjective and should not enter into the business part of this transaction. You could tell us (the caregivers on this forum) how you feel.
Then separate your feelings from the business end and tell them something like:
"I cannot possibly do that". "I need to re-negotiate the advancing money part, and will no longer be able to advance any funds to care for your loved one".
"This is highly irregular, you need to provide the funds to care for your elder or shop for him/her yourself"
"I understand how this arrangement is more convenient for you, however I need the money you pay me to live on, and will no longer use my funds to care for your elder".
"No, I cannot possibly do that"
"Can you provide the funds to shop for (insert name here) in a pre-paid card since we now have an idea what to expect monthly."
etc. etc.
So, now how do you feel about being exploited by your employer?

This really isn't about "feelings" -- it is a straight-forward business dealing. You should not be responsible for using your own money and then being reimbursed.

Don't get emotional with the client, and don't talk about how you feel. Talk about what needs to happen, and what has happened in other households you have worked in.

If they are not giving you money to run the household, then I also wonder what else they aren't doing. Are they withholding taxes for you, or do they make you consider yourself an independent contractor? Are you on duty 24 hours/day? (Thought that wasn't allowed any longer?) Are you paid under the table?

Goodness.

You don't complain, but for myself I'm shocked. Shocked that a family hiring a professional caregiver could even begin to think it appropriate to expect the caregiver to advance money to her client and then request reimbursement.

Boy, have they put the cart before the horse!

They should provide you with a credit card and/or a petty cash fund for your client's upkeep. You then provide them with a detailed monthly account, with receipts. It works that way round. Your client's family has got it back to front.

Not one cent should come out of your pocket to pay for your client's needs. No, not even if they pay you back. No, no way is that acceptable.

Happily, you don't sound nearly as indignant as I feel about it so you will be able to explain to them nicely how things are done.

How about this. " In my previous live-in caregiving jobs, i was given a certain amout of money which i used to run the home each month, up front. I provided receipts at the end of each month. Doing it that way avoids me running short of my own funds during he course of each month. Could we handle the house money that way?".