-Yes. You need to leave U.S. as soon as possible to get an H1 visa and come back just before your H1 becomes effective.
Is there a problem if I'd choose to stay at the US? Will it give me a trouble for my future status petition such as green card?
_When you violated your F1 status rule, you were considered out of status in March. When your unlawful presence > 180 days, you are not allowed to adjust your status to permenant residency (green card). Also, when you go out of U.S., you will not allowed to re-enter U.S. within 3 years or 10 years. This is very serious, so I suggest you to leave U.S. now and get an H1 visa to come back in September.