Substituting grease for oil in a gearbox was always the bodgers trick for solving oil leak and noise problems from wear. Triumph were quite firm on insisting heavy oil and grease should not be used in these boxes as it relies on splash lubrication to get oil everywhere around it, especially to the footchange and kickstart mechanisms.
I have various bikes fitted with Sturmey Archer boxes where grease is the recommended lubricant. Even so, they also recommended adding a teaspoonful or two of oil as well, then tilting the bike to the left with the engine running so the oil could get down along the mainshaft and in between the mainshaft and sleeve gear. They didn't trust grease entirely.
Albion boxes are the opposite, always lubricated with oil, no matter how old, but I regularly come across examples filled with grease. Invariably they leak like crazy when it is replaced with the correct oil. A proper overhaul with new bearings and bushes fitted sorts them out and is a much more appropriate action than hiding the problem with grease.
As I see it both gearboxes sound in need of some TLC. Keep riding your '59 5TA, overhaul the 5T box and then once it's right and on the road use the knowledge gained to sort out the 5TA box.
A nicely sorted box makes riding so much more pleasant. It mightn't be flash like new paint or chrome work but its much more satisfying.
Cheers,
