lease assignment questions

[WMargulies]

I'm in a rent stabilized apartment in Manhattan, with 4 months more on the lease.
I've taken a new apartment to start June 1 and am hoping to assign my lease to a new renter starting asap after June 1 as possible. I am aware that this person's name and various information has to be given to my LL in writing via specific methods. Once that is done, I'm not sure what happens next. Does anyone have any knowledge of or experience with this process?
How likely is my LL to reject a viable new renter? If they accept the proposed assignee - does that person's NEW lease begin on the day of my vacating the apartment, or does it have to be 30 days from notice, regardless? I'm not quite sure how to explain 'what will happen' to the people who are interested in taking the apartment over.
A second question is this - I know that we are required to give 30 day notice in writing. I also know that the LL won't show the place to find a new tentant on their own until we vacate. Either way, I'm probably stuck with paying June rent one way or another. What I'm hoping to find out is this:
Can I simultaneously notify the LL that I'm vacating June 1 AND give them a prospective (or several prospective assignees) at the same time? In that case, if they reject all of my assignees - does my "30 days" still start at June 1 or at the date that they deny all potential renters I present them with?

This seemed so simple at first, and now I have NO idea what is going on. ANY and all help, advice, experience will be appreciated.
One last thing - is it just easier to tell them now that I'm leaving and hope they find a new tenant in the month from when we vacate to July 1? (In which case I just suck it up and pay June rent?)
HELP!

:confused: