How should I announce that I'm leaving?
Since your departure is known to and supported by the head of the department, there is no real issue here: you can talk about your departure in any way that seems most reasonable to you.  I don't think there is any canonically best way to do it.

In my (limited) direct experience, faculty who leave but do not retire often do so without an "official" announcement.  The exceptions have occurred when we want to rehire into the position before the new academic year.  Maybe it would be better in some ways if there were an official announcement.  If you feel that way, I would talk to your chair about it and ask if they would like to make an announcement on your behalf.  


  What do you say to students?


That surely depends entirely on what kind of students you have and the nature of your relationship with them, about which I know nothing and you know everything.  I presume that as a PT lecturer you have not been advising graduate students in any kind of official capacity.  But if you know of specific students for which your departure would cause a definite and substantial disruption -- well, sure, I think you should tell them as soon as you know you're leaving.  That seems to be essentially common sense.  

Good luck in your new job.