You cant just get I-20 from school when you are in H1 visa.Once you are in h1 your SEVIS record is already cancelled so inorder toget new I-20 the school has to generate new SEVIS record for you for which you have to apply change of status by filling form I-539 along with the supporting documents like an essay mentioning why you want to change status to f1,  $300 fees plus there are a list of documents which INS/DHS requires. I have the requirement list print out I can email you if you want.

So pedro pls dont fall into false impression only changing between schools in f1 require new I-20. but change of status from h1 to f1 is completely different process. I know this because I have similar situation like yours I also went to my international advisor and asked for new i-20 as i didnt want to stay in h1 anymore. My advisor smiled and replied me back explaining school cant just issue an I-20 for me as my SEVIS record has already been canceled. I've to file change of status which takes 2-4 mnths and go thru the procedure. I dont wanna make things complicated but once again please contact lawyer for safe and sound solutions .

Remember, making sure that you dont loose your h1 status just in case your f1 transfer request gets rejected should be your top priority and lawyers are the only one who can give you proper advise by studying yor case in depth. BETTER SAFE THAN SORRY