Am I responsible for my mother with a Power of Attorney if she gets sick?

Only if you have Medical POA.

Florida

I think it would be helpful if we know in what state
GiGi11 was in, Many situations that vary from state to state do not
specify what they are in

Yes, you are responsible to get her to medical care, if you have medical POA and she is incapable of making decisions on her own (dementia or unconscious). You are not responsible for the medical bills. They should be paid by her insurance and out of her funds. If she doesn't have the means, then SSI/Medicaid should help. Contact the hospital's social worker for help applying. If you have financial POA, you are responsible for getting her bills paid and should have access to her money. This can be with a joint bank account, a revocable trust or as her SS payee.

In our state there's a medical power of attorney, separate from the day-to-day authorization. A suggestion for you is to find the legal aid office near you (free or with fees scaled based on income) and ask them.