I had an accident. I hit someone from the back. My car slid on wet road. I have full coverage on my car but my insurance say because I am at fault they won't pay for my injuries. Are they right?

In the US types of car insurance and what is covered varies greatly by state.

In GA for example, you are not covered for your medical bills unless you purchase additional coverage for medpay. "Full coverage" is a general term that means your coverage pays for the repair of your car. Issues like car rental, your medical bills and other payments will vary by what you purchase. So look at your policy page.

What is the company? Was there a police report?