Your employer Must provide you with a corrected W-2 once you notify them of the problem. It's possible that they too need to correct that amounts submitted to each state on your behalf.
I am unaware of any reason they would be responsible for "any mess" as you are not late filing yet.
If you filed incorrectly, you only need to file an amended return(s) once you know the correct amount of taxes that should be allocated and reported to each state.
If you haven't filed and it gets close to the filing deadline and you are still awaiting corrected W-2 s, you can always estimate what you might owe, if anything (so no interest/penalties accumulate) and use Turbo Tax to file the necessary extensions for the effected state returns until you actually receive the corrected W-2 s.
Again, I am not a tax expert or attorney but I am unaware of any liability on your employer's part of an error that they are notified of and correct in a timely manner.
* Just as an aside from working where W-2s and 1099s were prepared; they generally reflect the accumulation of what you've been receiving on your pay stub. Try to watch those especially as you enter the last quarter of the calendar year so you can notify your employer of any issues with information and taxes collected and disbursed on your behalf.
