Finally skimmed through it. There's no apply chart, but pages 73-81 give you the info you need to open up a spreadsheet and make your own apply chart. That said, the devil's in the details.
If the transmission makes the 1-2 shift, the governor is likely not the problem.
According to the apply chart I made, the 2-4 servo shifts the transmission into second, but isn't required for third gear. BUT! If you read page 45, the 2-4 servo has to receive release pressure to allow the shift into 3rd.
So, it could be your servo. It could be your 2-3 shift valvetrain. It could be your 3-4 clutch.
If you're thinking about swapping in a Corvette servo and your fluid looks good, it might be worth a shot. If you want to rebuild, or you've been towing a lot, or you have trash in your fluid, rebuild it.
** NOTE **
I can read the manual, but I've only ever rebuilt one TH700R4, and that was in a transmission class.