Since you are moving out-of-state.............you can not stay longer than you have to.

Yes they can ask and yes, you can say "no". They have zero legal standing in this case but they are certainly free to say whatever they like to encourage you to stay.

Not sure what you mean by "no contract". If there is no contract then they would have no legal obligation to pay you when you did the work.

Anyway, the answer about giving courtesy notice and being asked to stay revolves around how valuable the company deems your work. Clearly, if they ask you to stay and offer incentives, you get to decide whether or not to agree to it.  It's a free country.

I once quit an engineering job with 2-days' notice, to start another job, and the company immediately offered to hire me back as a consultant at double my prior salary, seeing that I was in the middle of 25 different design projects.  I happily did both jobs for about 6 months, sometimes working 24-hour days, until my new job wanted me to move to a different continent and working remotely for the other company became too difficult.  The "new job" only lasted a year and I ended up finding a much better job based upon my entire resume and glowing references from everyone who found my employment had been helpful to them.

You can't please everyone, but you can at least try to be a bit flexible when it's your reputation or career on the line.

You have no obligation to stay longer.  Find out if they'll make it worth your while.

Absent a contract to the contrary, you can legally inform them on the way out the door you quit, effective immediately. giving notice is legally a COURTESY. 

of course you can ... you're free, aren't you?

Of course. That said, if you gave inordinately short notice, anything less than two weeks being considered inordinately short, then if you want a good reference from your employer, you may want to give serious consideration to staying on long enough to give your employer those minimum two weeks that are generally expected, if at all possible. If you've given at least two weeks notice, however, then no one can fault you for refusing your employer's request to stay longer and you can leave guilt-free and with confidence that your refusal won't negatively impact any reference that employer might provide. 

For more technical jobs where finding a qualified replacement will reasonably take longer than two weeks, especially if the employee needs to come on board and complete training to take over your duties by the time you leave, or for a job where you know you're the only one there that knows how to do it and so should expect that your employer will need to rely on you to train your replacement, anything shorter than a month's notice is generally considered to be inordinately short notice. 

You can indeed refuse if there is no contract. When there is no contract and it's just a normal job, the resignation period is a courtesy for your employer to be ready for your absence from the company and to try to hire a replacement. 

If when you put in your notice they want you to stay longer, you have the right to say "I'm sorry, but my plans are fixed and I'm leaving the state. I can not stay longer."

They have to accept that. 

And further to that...if they start being mean about it (piling work on you, being hostile to you) you have the right to say "@#$% this!" and leave right then and there. 

They can't make you stay.  Remember, don't burn any bridges behind you.  It's much better to leave a job on good terms.

No just leave. But it is always a better idea of a need you to stay a little bit of extra time to try to stay.