The question here is when your visa status legally changed from F to H ? To my understanding, it is changed when your petition is finally approved. I would suggest you consult your immigration attorney to verify. If the effective date is 1/7/2020, you are set. No need to contact your employer. For more information, see H-1B
However, if the date is October 2019, please see the followings:
I will suggest you communicate with your employer to resolve the issues. Employers who do not comply with the employment tax laws may be subject to criminal and civil sanctions for willfully failing to pay employment taxes. For now, you will still file your 2019 tax return as usual. If things need to be changed on the W-2, you can always amend later. I am attaching the following links for your references:
FICA withholding
Employer vs Employee responsibilities
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