This is correct information.
The directors are responsible for holding the Annual General Meeting and making sure that an election is held. This is part of the responsibility that you accepted when you agreed to serve on the board.
I have to say again, you should not have directed your letter of resignation to the property manager. It is the other board members, and ultimately the members of the HOA, who make decisions about electing board members.
It's as if your boss at work came to you and said, "Hey, I've decided not to be your boss anymore. You'll have to find a new boss."
Boss = board of directors. Employee = property manager.
The people who can hire a new boss = members of the HOA.
If you want out, send a letter to your fellow board members saying that you will not accept election to another term. And start organizing your next Annual General Meeting to ensure that there is a quorum present to elect new board members.
This is the reason I said, in my first post, that you were misinformed about what you'd taken on in agreeing to serve on the board in the first place. You have responsibilities and are answerable to the membership of the HOA. If you just stop doing the job and don't let the HOA members know that there's no one out there looking over the finances and making decisions, you could be held liable if the property management screws up or makes off with the HOA's money.