Pete,
I am not a mechanic, I just went through troubles with auto gearbox, have some observations and studied the topic on this forum. If your symptoms are connected to any specific malfunction and experts here will recognize it - they will chime in.
First of all - the car has to be in tune. Idle needs to be adjusted correctly, meaning all the parametrs like ignition timing, points, mixture need to be correct. It is all inteconnected with shifting gears in automatic as it is related to vacuum. Your brake booster needs to work correctly (no vacuum leaks).
I do not know if you know when the gearbox oil was changed in your car. Oil and filter change (if you have not done it) may be a good place to start. It will also give you opportunity to see and smell the oil you have.
How to check if the gearbox "management system" and how to adjust it you will find here:
https://www.sl113.org/wiki/Automatic/StartThe management system is three-way solenoid and the throttle switch that need to work together. Adjustments are about regulating oil pressure with a screw and a rod. The rod and the three-way solenoid can be seen through the opening in the center tunnel where the passenger's feet are. The second adjustment screw is for the reverse.
There are also numerous posts on the forum on this topic, e.g.
https://www.sl113.org/forums/index.php?topic=22622.msg161719#msg161719https://www.sl113.org/forums/index.php?topic=24794.msg177342#msg177342All these checks and adjustments are not very complicated. You need a pressure gauge and a banjo fitting. I made the tool for the adjustment of a piece of metal sheet. Just mind the pressure units.
https://www.sl113.org/forums/index.php?topic=22821.msg163115#msg163115Why is your speedo not working? Is there an issue with the speedo drive at the gear box?