Some schools will say they only require a letter for matriculation, so with those you could complete your secondary now, you could postpone for a short while for the rest if you want, for DO schools as long as you apply by Thanksgiving you should be fine. As for asking for a letter, yes asking after 1 or 2 days of shadowing is fine, many physicians will be expecting it. In fact, the anesthesiologist I shadowed offered to write me a letter after my first day shadowing him, and I didn't even ask lol.