I have to laugh at the knee jerk response of "become paid" as the panacea cure all for this problem. I can't tell you how to fix your problem without more information, but I can tell you that I have the exact opposite problem.
We are non-profit ALS and have been around since 1953. We are stacking up volunteers like cord wood. We permanently put a second truck in service 24/7 and we are operating PPB. Sadly we may even be turning good volunteers away because we just don't have the space.
Believe me, I am biased but I think our service is the envy of the area. We got here because of motivated leadership with an emphasis on training and community involvement.
You are the minority, by far. It's quite a sweet deal that your municipal/county administration has going.
(Former volunteer here, too)