Wait until you are laid off to visit your family. You're just getting really discouraged and burned out, worrying about the impending job loss, and that is very natural. But my best advice is to hang on a bit longer.
While you are drawing unemployment and looking for other work, you will be able to rest and clear your head, can get in the visiting, perhaps get in some counseling to help you deal with your stress.
At this point, having indicated that you are in one of the positions about to be let go, your employer may be hoping very strongly that you will go on and quit your job. Nicer employer, such as chyvan describes, who allowed the girl to be laid off and draw unemployment benefits when she was really about to quit for a disqualifying reason do exist, but employers are not always that way.
Sometimes they up the demands and treat you worse in hopes you'll just tuck tail and run to avoid the pressure. When you say that your job is becoming more and more challenging, that they're asking much more of you, and putting pressure on you, that very well could be because they are hoping to frustrate you into quitting your job.
Though this may involve changing your personality somewhat, try not to sweat it so much. That you wanted to help out your co workers shows that you are very invested in doing a good job and when you say you do much much more than your actual job involves, this leads me to believe that you are doing too much. How do you think this is going to be good for you? After all, you're going to lose your job anyway.
If you have more than you can do, or you are asked to do jobs for which you have not been adequately trained, just do your best, take your time, and don't get wildly stressed if they criticize your work or tell you that your work isn't up to par or you must do more or better or they may fire you. This is what they do when they're hoping you'll quit, so they won't have to lay you off. Just hang on, be quiet and polite (and don't broadcast that you're going to be laid off to anyone else in the company, per my original advice.) If criticized, just say, "I'm sorry. I am trying to do my best."
Remember, employers don't like to pay unemployment insurance benefits. They have some familiarity with them, and know that if you quit your job, you very likely will not get to draw benefits, and their tax rates won't be increased. They may think this new manager understands things a lot better than you realize, and may give him/her kudos if she can drive you out.
But also remember that they can ask you for more and more work, may criticize your work unfairly, can ask you to perform the impossible (could ask you to "spin straw into gold" as they say) but all you have to do is keep showing up, do your best and let them be the movers in your leaving. If your health becomes affected, or you need personal time off to deal with your or a close family member's illness, we can go into whether or not you might qualify for FMLA, but since as you tell us you have no vacation time accumulated that you could use, for any other reason it probably would be a case of, "If you want to go home, then you have quit!" and they might just win that argument in unemployment.
