The situation is the opposite of what you ( I think) have written above.
The only slip in an auto is in the TC and when it is slipping it is generating heat.
When the TC clutch engages, the drive becomes direct through the planetary gears in the gearbox and there is no heating of the fluid by the slipping action of the torque converter. .
Thus the transmission runs cooler.
You should always aim for TC lock up particularly if towing.
If you cannot get it in 5th then manually select 4th and there should be a speed where it locks.
However it is my understanding only from threads on this forum , that this is not the main problem with the transmission which is a GM transmission.
The main problem is that the valve body material is soft and is worn away by the fluid action with normal operation and that almost all fail before 160KK.
GM uses this transmission in the Commodore high output V6 and they have revised fluids many times trying to fix the problem.
There is a discussion here and if you google you will find a lot of commentary.
Auto gearbox problem
Regards Philip A