First of all, are you doing a premed committee packet? Where are these writers sending their letters? Where you have them finally sent depends on which application service you are doing. For AMCAS they can just send it to the AMCAS address listed on their website. Also I think they have a form that goes along with it so it wont get lost. Make sure to ask your writers to put it on letterhead, signed and sent with their signature along the flap (usually they do this anyways). They need to be sent after you submit your application, so maybe ask them to send it in on June 3rd, no later, no sooner.
However, the fact that you havent really started getting your letters ready is really scary... Writers hate getting short notice on things like this. I personally usually give at least 1.5-2 months notice. Ive heard many professors and doctors tell me how much they hate premeds rushing letters. My advice is if you need them in such a short time make sure to tell them Im sorry that this is such short notice but.. They will appreciate that Im sure. Also, maybe give them a self addressed envelope with a stamp on it, that way they have no excuses for it getting lost.