Servers suddenly unable to make new connections; seems like ephemeral port exhaustion

Sounds like your suspicion about the July update being the source of the problem has merit. Try uninstalling the July 11 2017 Rollup from one of your affected servers. If problem goes away, consider contacting MS support? Then again, as its a "known issue" that might not be very productive...

Known issues in this update KB4025341

Symptom: Due to a defect in WLDAP32.DLL, applications that perform LDAP referral chasing can consume too many dynamic TCP ports (potentially exhausting them)

Workaround: To work around the problem, restart services or applications that perform LDAP referral chasing to free TCP dynamic ports.

The script mentioned here might help diagnosing what process is using up the ports. Basically it combines the output of netsh int ipv4 show dynamicportrange tcp with netstat –ano –p tcp to help with the diagnosis. The text also mentions this is only for finding issues in user-mode processes only, explaining how WinDBG must be used for diagnosing issues in kernel-mode processes.

This thread also mentions Event 4231 as another indicator of the issue.