I discovered the Environment issue myself earlier but it's a good point to emphasize, as it's not intuitively obvious that it cannot be shared between threads.
Recently I've removed the process instance logging (JPAProcessInstanceDbLog) (it was too costly for performance and I did not absolutely need it) along with some other changes (including putting Spring transactions back in after trying to do without them for a while), and the concurrency issues seem much improved. At some point I'll try to go back and figure out whether it was the logging or some other change that made the difference.
Jon