I will look into this.
However, I find it very unlikely that multiple people are trying to sign up for the same spot at the exact same time.
You may want to investigate any caching plugins you have active on your site, and make sure that caching is DISABLED for the pages with the volunteer sheets. I think there may be a chance that the page is getting cached with the info showing spots still open, even though they have been filled already.
Otherwise, how often does this actually happen? The only way I can see it happening is that a bunch of people somehow have the volunteer page open in a tab for a long time, and walk away or come back to it later and then try to sign up without first reloading/refreshing the page.
I can certainly add another check and change the error message that is displayed. However any link they click, such as any of the “back” links, already refreshes the page. But, if you have caching enabled on that page, the info it shows may not be up to date, so make sure you don’t have any caching plugins set to cache that page.