Also hoping that she will be able to get what she put into her teacher retirement (It's required) back once she leaves and we can roll it into an IRA
Questions: How long has she been teaching?
How long would it take to be vested in her pension?
To give some stab-in-the-dark answers, if she's been working only 1-2 years and is pretty far from being vested, I'd suggest that she take the money and roll it into an IRA. They have to return any money that's been pulled from her paycheck.
On the other hand, if she could be vested at five years, and she's in her fourth year -- I'd say stay 'til she could finish five years, and leave the money in the pension plan. Obviously it'd be years 'til she could access the money, and she wouldn't have a full pension like a 30-year teacher, but she would have a small pension coming to her at some point in her life. IF you trust that your state's pension plan is solidly financially and will last. It's good to have multiple money-sources in retirement, so keep this in mind.
Yes, look at the contract she signed. She is most likely paid for 9 months of work, but spread it out over 12 months. If you are really worried about it you can quit August 1st instead of May 1st. If she quits May 1st, she might get a lump sum paycheck (3 months worth). If this happens, lots of taxes will be withheld that you won't get back until next February-March.
My contract states that I work ten months and receive ten paychecks during those ten months /no checks in the summer.
It sounds like your wife has opted to "voluntarily delay" a portion of her paycheck until the summer, allowing for equal checks throughout the year. However, she's EARNING that paycheck in the ten months she's working -- so she will receive those paychecks in the summer.
One more thought: The county office holds my professional teaching license. If your wife's the same, be SURE she retrieves her license when she leaves.
I'd also suggest that (if she's near the renewal point) she keep it active -- this does require continuing ed classes, but if she should decide to return to teaching, it's easier to renew than it is to get a brand-new license once yours has expired. For example, back when I got my license, I needed a college degree plus the NTE (National Teachers Exam) tests. Today the NTE is gone, and college seniors are all taking the Praxis test. If I allowed my license to expire, I'd have to go back and take the Praxis, which would cost me money and time.