By tuning I meant checking/adjusting the correct pressure in the gearbox.
If you press the brake and hear air noise, your booster may be leaking, which is affecting the vacuum in the manifold, which is affecting how gearbox operates. So probably the booster may need to be checked/fixed first.
How did the old oil look like? Smelled like?
I am not an expert - perhaps your gearbox is indeed up for overhaul and slipping on start may be a clear symptom of that - I am not sure.