The LCIII has 32 bit clean ROMs and therefore doesn't need Mode 32. It was the older 00, 020 and 030 processor machines that didn't have 32 bit clean ROMs that needed Mode 32 to enable 32 bit addressing.
Sounds to me like you're getting software errors.
I'd start from a clean fresh install of System 7 and switch off all the non essential extensions and control panels and work from there. The Sonnet extension needs to be in the extensions folder and the system enablers simply loose in the system folder. From memory you need to install the Sonnet software with extensions off, so install a fresh system, zap the PRAM, remove the non-essential extensions and control panels, boot with extensions off and install the Sonnet software. Reboot and if it still crashes, then either you still don't have the right Sonnet software and/or there may be a problem with the card itself.
If you don't want to overwrite your existing System Folder (and you probably don't if you you've got a lot of application preferences in there) simply create 2 new folders and move the System file into one, the finder into the other and rename the System folder itself as something else. When you now install the fresh System 7 it will create a new System Folder on the hard disk. It's now easy to go back your previous system by reversing the process.
Sounds to me like you're getting software errors.
I'd start from a clean fresh install of System 7 and switch off all the non essential extensions and control panels and work from there. The Sonnet extension needs to be in the extensions folder and the system enablers simply loose in the system folder. From memory you need to install the Sonnet software with extensions off, so install a fresh system, zap the PRAM, remove the non-essential extensions and control panels, boot with extensions off and install the Sonnet software. Reboot and if it still crashes, then either you still don't have the right Sonnet software and/or there may be a problem with the card itself.
If you don't want to overwrite your existing System Folder (and you probably don't if you you've got a lot of application preferences in there) simply create 2 new folders and move the System file into one, the finder into the other and rename the System folder itself as something else. When you now install the fresh System 7 it will create a new System Folder on the hard disk. It's now easy to go back your previous system by reversing the process.