Unfortunately, the only option you have at this point is to make adjustments to your W-4 to ensure enough federal and state taxes are withheld.

Keep in mind, you can change your number of payroll exemptions as often as you wish by simply submitting a new W-4 to your employer. However, while an employer can implement any changes immediately, it doesn't have to apply it until the start of the first payroll period ending 30 days or more after you submit the new W-4. In a practical sense, this could limit the number of possible changes during the year.

Here is the form if you want to print, complete, sign and return it to your employer: Form W-4

So what happens when they messed up your W-4, you tell them repeatedly they're not withholding enough, they don't fix it, and this goes on for about 6 months. Then tax time comes and you now owe several thousand dollars? 

As far as your tax return goes the answer is the same as above - you have to fill out your taxes and take the hit.  The taxes that you are paying at the end of the year are the same that you would have paid if the withholding had been done correctly, just now all at once.  The IRS expects to get paid one way or another.

As far as what to do about your employer you can report them to the IRS for failing to withhold correctly.  Especially if you can document your back and forth with them about the correct amount you requested be deducted.  If they are doing it to you they may very well be doing it to other employees.  Here is where to report an employer.

In addition, you may have reason to report your employer to the authorities in your state.  You'll have to check the laws in your state about that.

@TheWhiteMocha