Your employer's lawyer may be wrong. You do not need to leave based on the information you provided. You are eligible for COS as long as you maintained you H-4 properly. COS does not require visa stamping; that is the whole point of COS. There may however be another factor involved, which you are not aware of. The lawyer may also have filed the paperwork for consular processing by mistake instead of COS. I'd talk to your employer and their attorney to get better info. You should not need to leave the US at all.
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Whether you need to leave depends on whether USCIS changes your status or not.
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No, if your H4 runs to the day when H1B employment starts.
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