I manage 30 - 35 managers depending on the time of year. In my experience it sounds like your supervisor is over her head and is stressed out. Rather than asking her supervisor for help (if it is available to her) she lashed out at one of the people on her team willing to help her out.
I am not sure what the laws in Iowa are (read your location under your avatar) but in California you are not obligated to sign any write up. Your supervisor will still file it with a "refused to sign" note attached. It will be used in the event they need back up to prove that a given situation was addressed.
If you do not sign the write up and request your file at any time they are not obligated to provide that document to you though.
The whole thing sounds unfair and the result of under trained managers. You should put a call or email into your Human Resource department. I recommend email at some point so that you have written proof that YOU reached out to make the situation better.
Don't attack the manager, ask what the company policies are on scheduling and what their policies are on the schedule requirements of your particular position are. Since you are an established employee that is being "asked" to help out sounds to me like you are in the right to refuse the extra hours.
In my world we have a lot of students as crew throughout the company and do our best to work with their schedules. Not to abide by laws, but to keep happy and hardworking people on our staff. If you have a decent relationship with your manager simply ask her why she wrote you up. Let her know you want documentation of the companies scheduling policies and why she thinks you are being insubordinate because you are not willing to adjust your personal life around a job that has suddenly changed the expectations on you. If not go directly to HR.
I would think that signing the write up with the notation or not, will let your manager go on believing that you agree at some level and she will continue to not respect your scheduling needs. Finding a new job is tough right now and on the other side of the coin no company wants to think about labor related law suits either.
Good luck.
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