The president has no unilateral power to force the dissolution of a nonprofit corporation.  It would have to be approved by the board of directors, and if there are voting members, probably by the members as well. 

A vacancy in the office of president does not automatically cause a dissolution.  Normally a vice president or some other individual is authorized to handle the duties of the president in the absence of the president.  In the long run, you need someone willing to take the leadership role.  But in the interim you might be able to entice several people to serve as co-presidents, or let an executive committee of the board or the board acting as a committee of the whole run the organization.  It sounds as though you might be better off with no president than with the president you have now.

I would be very wary of his effort to give your fundraisers to his other organization so that he could run them.  This sounds like blatant self-interest that he is seeking to advance by peddling false information.  It raises concern about everything he says. 

Even if you decide to dissolve because there isn’t anyone willing to provide leadership, I would not want to turn the fundraisers over to his other organization if it inures to his personal advantage.  I would look for other organizations to help, or would negotiate with his other organization to assure that he receives no personal benefit.  It doesn’t sound like he is offering you many options, but this is a group decision that you ought to make on the basis of accurate and complete information, not his self-serving false statements.