Hi, The first thing to consider is if both the gearbox and the overdrive are original to your car. Do you know for certain that they are?
The reason I ask is that the basic gearbox was used by Triumph in a number of cars and there are many, many variations. There were at least two different A-type ODs, too. Your OD should be the later one, with smaller accummulator piston that makes the OD engage a little more softly. The TR4 and earlier A-type used a larger piston that tends to "bang" the OD into action. In fact, for hard driving, the IRS cars really should have upgraded rear drivetrain to handle the earlier type OD.
Assuming both gearbox and OD are original to your car, that might help simplify the job and finding parts, to a degree. You'll need a good, correct factory manual, plus the OD supplementary manual would be helpful.
To work on the gearbox, there are at minimum two special tools you'll need to buy. The Roadster Factory sells them for about $100 apiece. These are almost essential to taking the gears off the mainshaft, it's an almost impossible job without them.
According the manuals, you'll need 14 additional special tools to work on the OD. However, I'm sure some could be made up or worked around. However, others will be mandatory. I know a high pressure gauge is necessary, for example. The A-type operates at up to 400 lbs. psi.
There are some upgrades possible to both the gearbox and the OD unit. These include things like roller bearings to replace bushings, roller-type thrust bearings and stronger gears or shafts. I don't know all the details. You might inquire at some of the British specialists.
What I'm leading up to is that I have no plans to work on my own overdrive, which will be rebuilt before use. I'm relatively mechanically inclined and enjoy working on cars and motorcycles, have built more than a few engines and some gearboxes, even worked in the service dept. of an auto dealer when I was much younger. But, the OD is a specialty just in itself. I'd rather have an expert who knows these things inside and out and has all the necessary tools and test equipment to do the work, do it once and do it right. That's my opinion. I'm still looking for that A-type "expert" though!
There are some tests you can do to the OD unit, to help find out if the slippage is due to a worn clutch, softened springs or too-low pressure. First check that the filters are clean, that the actuating valve is clear and the oil is correct and fresh.
I think you would be wise to rebuild (or have rebuilt) both the gearbox and the OD at the same time, since they share oil and some of the residue from the gearbox can end up in the OD, and vice versa. At least, if one is rebuilt, the other should be thoroughly cleaned before putting them back together and using them.
Happy holidays!