The currently available Service Manual (SM 1002-R6) lists MIL-L-2105B (I looked this up and as best as I can determine it is equivalent to API GL-4) or just GL-4 for most gear applications. Going beyond GL-4, ie. GL-5 etc. puts the shear additive too high to be safe for yellow metals.
This does not take away from the accuracy of the earlier Willys manuals that call for straight mineral oil or GL-1 etc.
I simplified the number of gear oils I needed to stock by using GL-4 85W-90 for all the gears, including differentials, since that gear oil is safe for everything. The only deviation is I used GL-4 140w in the knuckles (SM states GL-3 but the GL-4 is what I could find) since my axles had been changed. This gear oil is specified in SM 1046 where all knuckles were now Spicer (cardan cross) vs. the CV versions in the earlier jeeps.
So, after more words than probably necessary yes, the same oil in the steering gear as the transmission should be okay.
AFAIK, changing the gear oil in your transmission won't affect the leak. However, if you have no idea how long that oil has been in there, changing it could be a wise move. Eventually, when you have time, separating the transmission from the bell housing to see exactly what is leaking would be good.
Hang on to your steering adjustment screw/lock nut (keep it safe) as you may or may not want to go back to it after running the tight steer.