I think the answer above pretty much nails it. Our road association went through a similar upheaval in 2010 - 2012, but things worked themselves out eventually. What happened with us was that there had been a non-profit road association that went through essentially what your association is going through now.
Those who had served for a number of years with few signs of appreciation from the rest of the members eventually burned out and quit, one after another. Only about 1/3 of the members paid their dues. Our daughter had recently moved in, and my husband and I had not yet bought our property there. One day the President swung by my daughter's house and dropped off the files and said, "I'm moving out of state. You'll make a great president."
The following year they were unable to raise enough funds to pay the plow contractor, and word quickly got around so no contractor would bid on the job. One of the members agreed to plow the following winter, and a few of the members paid their dues a year in advance to get him paid. But it was obvious things could not continue that way.
In 2012 we voted to scrap the non-profit association and start over as a statutory association. We now have almost everyone paid in full! But we also struggle with finding people who are willing to take on the administrative duties, and burnout is on the horizon for some of those who have served for several years.
Perhaps if you share this story with your members and point out the possible consequences of having no road association, someone will step up to fill the gap. Also, if some of your members are moving out, maybe you'll get some new blood in.
One thing I would recommend is to find some way for the members to express appreciation for those who serve. And try to make meetings fun. Have a potluck or hand out door prizes or funny awards. One of the things that killed our original non-profit association was that annual meeting always turned into a blame fest, with everyone yelling at each other. After a couple of years of that, hardly anyone would show up.
Also, try to bring new people on board in an apprentice role. That way they can ease the burden on the officers by taking on some of the responsibilities, while at the same time learning the ropes.