61 American convertable.

Easiest to change the entire axle. What engine and trans do you have? IIRC you mentioned 196 OHV and an auto trans? A 3.31 axle would be good in that case -- I ran that combo for 14 years. Tried a 3.78 axle once, did nothing for performance except increase engine rpm at speed. I timed it once and I think it cut about a half second off 1/4 mile time (not real accurate, just marked out about 1/4 mile on a back road and used a digital watch with stop watch function). When it takes 16 seconds or so a half second (or even a whole) isn't noticeable enough to change the axle, not with the extra 500-600 rpm (and loss of about 1 mpg) at cruising speed. The "butt dyno" didn't notice a thing, that was the clincher!

The best thing to do is LEAVE THE BIG NUT ALONE!!! Disassemble the u-joint under the car. Raise the back and put jack stands under the axle (weight has to be on the axle so the axle swings back a bit on the springs). Then slide under and pop the internal c-clips off the u-joint caps of the yoke on the axle. Then take the shaft out. I use a wood block against the very end of the shaft to knock the shaft sideways to get the caps out, BUT BE CAREFUL! The yoke welded on the end of the shaft extends about 1/2" up into the tube. MAKE SURE your wood block is on the end yoke, NOT the tubular shaft, or you will dent the shaft and throw it off balance.

Easier said than done, but taking the yoke off the axle by loosening the big nut can be a problem. For starters the big nut is tightened to 250 ft/lbs. If you're using a lift it's not too hard to get off and tighten with long wrenches (two needed). I've done it on jack stands, but I was a lot younger! The yoke has to be a certain measurement from the front of the axle.. I think it's 3-3/16" from front of axle housing to center of u-joint cap, but check a TSM to verify that measurement. It's just a lot easier to leave it on. I've done it both ways, and I won't take the nut loose again unless I have to change the yoke for some reason. Even with the car on a lift you need to put a trans jack or something under the rear axle to lift it, as hanging down the springs pull the axle forward and bind the shaft between trans and rear axle. Of course you could take the springs loose from the front, but that's a good bit more work, and you still need to lift the axle to get them loose.