Rarely anything is automatic in US immigration law. You do not need I-20 if you continue maintaining your H1B status including working for the employer on the conditions outlined in the LCA and going to school in the spare time. If you need to seek change of status to F1, you need to do that before you quit your job.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter,not should it be viewed as establishing an attorney client relationship of any kind.
No I-20 will not automatically cancel your legal H1B. You need to prepare and submit all required F1 documents and get them approved BEFORE your current status expires however.
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You only need to stop working once your COS F-1 is approved.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.