504 errors on Jetpack/Contact Form 7 submits after 4.6.9 update

I get 504 errors when I try to submit forms on these pages:

https://thrownstone.org/about/contact (jetpack form)
https://thrownstone.org/participate/partner-with-us/ (Jetpack form)
https://thrownstone.org/participate/volunteer/ (Jetpack form)
https://thrownstone.org/participate/playwrights-actors/ (contact form 7)

But no problem submitting this form:
https://thrownstone.org/participate/sign-up/ (Mailchimp for WP plugin)

Deactivating plugins to test for conflicts, it appears The Events Calendar and The Events Calendar Pro are the culprit, but downgrading from 4.6.9 to 4.6.8 was disastrous, and I had to restore a site backup. I’m losing a lot of form submits, so this is fairly urgent.