I was involved in a low speed car accident while leaving a parking spot, can I face criminal or civil liability for leaving

Yes, you can face both civil and criminal liability for this in Texas.

Have you been sued or accused of any crime?

Do you know who the owner of the car was, or any way to figure that out now if you don't know?

I'm not sure what you're looking for here. I'm not 100% sure what you mean when you say you "kissed a car" -- I assume you mean "touched lightly with your car," but maybe not.

Perhaps you can re-word your question or provide some further details that will, in turn, help people here give you more useful answers.

You face both civil and criminal liability. You should report the loss to your carrier immediately and try to locate the car's owner and ask them if they suffered any property damage.