Thanks to Kevin (kb2XXX) for helping me debug some comms issues - I think I can talk with fldigi now.
Quick background - I run my rig using a Virtual Machine. Host is running CentOS 7, guest is Windows 10. That Winders machine is what all my control and logging software runs on.
It looks like a recent CentOS system update has caused a problem with Hamlib (which WSJT-X and JS8Call use) on that guest VM. Not sure how to fix it, but at least I think I've isolated the issue.