So the way I found to do this — short of getting a pre-Sierra Mac and short of asking Omni for help — was to set up a pre-Sierra virtual machine on my own Mac.
Here's the short version:
- Install VMWare Fusion. There's a free version if you're using it for personal use.
- Get a version of the full InstallMacOSX.dmg for Max OS X El Capitan 10.11.6 and extract Install OS X El Capitan.app (see more below). The version of El Capitan on Apple's "old versions of macOS" webpage is just an updater, so you'll need to find a complete one online or locate an original.
- Create a new virtual machine using Mac OS X El Capitan.
- While that virtual machine is running, select Virtual Machine > Install VMware Tools and restart the virtual machine. Once restarted, these tools will let you drag and drop files from the host machine to the virtual machine.
- Download OmniOutliner 3.10.6 from Omni's downloads page.
- Drag and drop from the host machine to the virtual machine the OmniOutliner 3.10.6 installer and any files you want to convert.
- In the virtual machine, install OmniOutliner 3.10.6. It's got a free 14-day trial so you won't need to dig up ancient registration codes.
- Open each file to covert and click Upgrade when prompted. This will save a converted .oo3 file in the same folder where the .ooutline file is.
- Drag and drop the new .oo3 files back to the host computer.
- You can now open those converted .oo3 files in recent versions of OmniOutliner.
The longer version involved a huge detour on #2 above. VMware cannot use InstallMacOSX.dmg directly and, per user dempson on this post in VMware's forums, recent Macs won't let you extract Install OS X El Capitan.app if your computer model isn't within the OS's range of acceptable models. The workaround… create a virtual machine of your current operating system (since there'll be no hardware check) and extract the older operating system there. Pretty convoluted, but it worked for me.
Here's the step-by-step, adapting the solution provided by dempson for this scenario:
Download the installer for the same version of macOS you are running, e.g., via App Store or one of Apple's support pages for downloading recent macOS versions (which does the download via System Preferences for macOS 10.13 High Sierra and later). This places an Install macOS [version name].app into your Applications folder.
Create a new macOS VM in VMware Fusion for the version you've just downloaded. Point it to the installer application.
Once the VM is created, proceed with installation of the recent macOS into the VM.
Once the installation is complete, go through initial setup.
Install VMware Tools in the VM and restart the VM.
Drag InstallMacOSX.dmg for Max OS X El Capitan 10.11.6 from the host into the VM.
Inside the VM, open the .dmg file and open the installer package. Proceed with installation.
Once the installer completes, you will have Install OS X El Capitan.app in the VM's Application folder. Drag that application back to the host.
You now have an install application which directly usable by VMware Fusion to create an El Capitan VM.
I also had an issue where El Capitan wasn't recognizing my MacBook Pro's trackpad or my mouse and got hung up at a picture of a mouse during the installation on the VM, as described here. The solution for that was to select Virtual Machine > Shut Down, select Virtual Machine > USB & Bluetooth > USB & Bluetooth Settings… and to downgrade USB Compatibility to USB 2.0 under Advanced USB options, and then to turn the VM back on, as described here. Not sure if that was specific to my hardware set up or will happen to everyone.