The availability for part-time vs full-time is a state thing. I'd say the norm is that you must be available for full-time, but that only means that if offered full-time, saying that solely because it was full-time would not be good cause to refuse it. On the flip side, the state rarely gets involved in the type of work you apply for. You're more than welcome to confine your search for part-time work on a shift that you perfer. Just remember that because you apply for what you want doesn't mean that you won't be offered something else, and that still means that if you turn it down, there's going to be a refusal of work issue to decide.

There are some state's that if you work the correct amount of time in part-time work, and lose that part-time work, that you may limit your search to part-time work and refuse offers of full-time work for that reason because your prior employment was part-time.