I have to point out that our Constitution is a little contradictory on the matter. It says on resignation a new committee member must be offered for election at next meeting.
It also says how an election process has to be run. Including a 8 week lead in process. But it doesn't address the conflict between these two rules... Which makes it tricky. There are other odd bits and pieces as well.
I think what we have done is a reasonable compromise but there are many compromises to be made.
I'd suggest if you want to help us fix it please join us