No, you cannot submit your application without the personal statement. However, you can submit it without the LOR's (of which there are 4 slots) as long as you already filled out the writers' information.
You should submit it as soon as you finish filling it out (including the PS and LOR writer information).
Most schools are on a rolling basis and will only offer you supplementals and interviews once everything is received: DAT scores, LORs, personal statement, transcripts. Some schools will ask you to pay them a secondary fee regardless of your AADSAS status.