No. Your employer will correct the employer payroll reports and issue you a corrected W-2 for your social security and medicare taxes (commonly known as FICA) paid by you. This will correct the Social Security Administration records for you as well.
The FICA taxes will not affect your tax return since the correction is coming from your employer. You can file your return with the correct amounts shown on your W-2 (confirm with your employer), adding the correct social security and medicare taxes, as they appear right now, in the appropriate boxes .
There is no need to wait for the W-2C before filing. The only boxes that will change on the W-2C should be 3, 4, 5 & 6 (maybe only 4 & 6).
[Edited: 03/16/2021 | 5:33a PST]
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