I would get ready to make attempts (I was living in my car at the time) and would end up burning all my important documents, burn checkbooks, birth certificate, etc and cut up all my ID cards inciuding my retiree ID (I was medically discharged from the Marines so they consider me temporarily retired due to disability). I was living in my car and had all my documents in the car. I was extremely paranoid while living in my car with schizoaffective disorder. I didn't want anyone knowing who I was and didn't want my docs getting stolen if someone broke into my car at some point as well. I went through this purging ritual at LEAST 2 times. Cut and burned every document I owned. My birth certificate, military documents like my DD-214, my checkbooks, my Social Security card, registration, everything and donated some of my clothes, shoes, and blankets. I was that paranoid about being identified. Maybe it was because my car was my home and I didn't want people knowing I lived in my car because I believed I could get in trouble. I definetly regret it now. But at the time it felt like a cleansing and identity erasure. I've got an apartment now and have calmed down a bit.
TL;DR
But my advise: they're going to identify you anyway by fingerprint or dental. Or maybe not if you haven't had any run ins with the law or been to the dentist, ever. One of the gifts of being dead is that you don't have to care about who finds out. But maybe drive to another city if that makes you feel better. Drive out to the middle of nowhere, even better. But keep your keys, IDs, wallet etc in a waterproof case nearby. The police would appreciate if your belongings were not covered in dead body smells and juices. So like a plastic toolbox near your body: inside the toolbox your keys, IDs, wallet, etc inside a ziplock bag. Or keep your wallet, IDs, etc in your car. If you go out to the country or middle of nowhere you have less to worry about having your things stolen off you while unconscious.