Firstly, the EC is obliged to try to find volunteers to fill the vacant posts until the next AGM so that might solve the vacancy position (if you can find a volunteer and the two remaining EC members can agree).
A better option might be to find someone to take the retiring member’s place as a proxy – ie, the member doesn’t resign until the AGM – and carry on until you can sort this out at the AGM, giving you time to identify some potential new members.
At the AGM you first have to decide on how many members there will be on the EC (maximum 9, minimum 1, answering your other question) and you’d do that on the basis of how many people volunteered (or that number minus 1, so you can vote the troublesome member out).