“Which state is wrong?”----> NJ employer is wrong; on Form W2 line #15, state; NJ and your employer in NJ should put his/her EIN on line #15. You need to contact your NJ employer.
“What will happen if I file my taxes exactly as is, without any corrections? (Will I wind up paying more/less or being penalized for their mistakes?)”----> Then as you’d overpay NY( if your NJ employer didn’t withhold taxes from your NJ paychecks, I mean), you will get a refund of taxes paid to NY. Since you did not have NJ withholding as said, it is possible that you will owe taxes to NJ. Your employer must do one of the following; he needs to issue a new, corrected Form W-2 to you that includes the correct state name. and indicate “REISSUED STATEMENT” on the new copies, he should issue a Form W-2c to you showing the correct address in box 15 and all other correct information, He should reissue a Form W-2 with the incorrect info. To you in an envelope showing the correct info, or otherwise deliver it to you. Your NJ employer may be subject to a penalty as provided under section 6721. The penalty applies if he includes incorrect information on Form W-2. So, you need to contact HR or payroll dept or your employer.
Please visit the website for W2C here; http://www.irs.gov/pub/irs-pdf/iw2cw3c.pdf
“or being penalized for their mistakes?”---->As long as your NJ employer doesn’t withhold taxes( state) from your paychecks, your NJ your employer is liable to the taxing authorities for withholding when it is required, not to you. However, you would file your NJ state return based upon your actual income along with interest and penalties owed by not withholding taxes( I guess you report them to the Internal Revenue Service; you can not get the NJ employer to pay your taxes for you, even if he didn't withhold taxes as required). Your home state is NY and you earned income in CT and NJ as nonresident of CT and NJ, so you pay your state income taxes to CT and NJ and can claim your state tax credits on your NY state income tax return.