I would say you have a dead solenoid, but that is not true with this transmission. It does not have a Reverse solenoid.
Reverse works by pulsing one of the regulation solenoids to get reverse. It is a very unique process for this transmission.
Question? Is it the same effect in reverse with the switch in S or W? As these are two different ratios, use different friction packs too.
By looking at the logic table of this transmission. B3 is only unique to reverse, and works fine cold, but hot fails to work? It sounds like the problem is all valve body, but it could be a worn clutch pack too. Manual Valve is leaking(the valve that takes our shifts in the valve body. See attachment. See clutch application.
See Condition on the top of last page of attachment.. It could be low of fluid. Have the 722.6 testing dipstick? Buy a 722.6 dipstick on ebay?
Wrong fluid? How old is the fluid?
Lastly, It could be just flat-out worn-out. I wrote a thread on how to determine the life left on a 722.6. It works for any year 722.6 equipped car.
One more, 96 was a bad year for the 722.6 to early 97 many flaws for the V8 & V12 cars...96-97 are only compatible with themselves. 98 to present(722.6 is still being made) are compatible with each other.