Yes. If the employer will not correct the situation you should file your return, then send in Form 843 with your two W-2s and a letter from the employer.
The employer will be liable for the repayment to the IRS and the IRS will handle that part of it. When an employer files their employer payroll tax report (Form 941) a math formula is used which would not allow them to pay more than they owe based on the wages they paid. It is for this reason that when only one employer is involved they actually have the tax dollars and it was never paid to the IRS.
For your convenience as indicated by @DawnC the information is posted here again.
To get a refund for the excess withholding, fill out IRS Form 843: Claim for Refund and Request for Abatement according to the Form 843 Instructions and mail it in separately. Make a copy to keep with your tax return paperwork. It is a pretty straightforward form, but if you have any questions, you can reach back out here and someone can help. You will be claiming the same amount you adjusted the box 4 amount above which is the amount of social security tax you overpaid
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