Car Insurance Question for Young Graduate

I am graduating in a few weeks and I will be moving back home with my parents for at least 6 months. Since I will be at their household will I be able to remain on their policy? The car is titled in my name (which the agent apparently is unaware of-he is pretty worthless) so should I add my dad onto the title? If so, how would I do that and how much would that cost? My mom is freaking out and thinks they would not cover me if something was to happen since the car is in my name. I live in Texas if that matters. I did do a little research but it wasn't very clear. Thanks for the help!