More than likely your trans external linkage is adjusted properly from what you mentioned.
It's tough to diagnose problems over the internet, can you share some more insight as to what you mean by better traction? Does the car spin the tires when you take off from a stop in D but not in 1 or vice versa?
Are there any other symptoms when the transmission is cold?
I added some extra info -
Make sure to check the trans fluid level in neutral, engine running with the car parked on a level area. Ideally you would want the fluid level to be near the top of the crosshatch area once you have driven the car for maybe ten minutes prior. Over is not good, likewise, lower than the crosshatch area is not good.
With the engine off, make sure that the kickdown is adjusted so that wide open throttle on the carburetor also equals the kickdown linkage is pushing the little lever on the transmission as far back as it can go. Assuming you have the stock kickdown linkage, disconnect the kickdown link rod from the carb, open the carb up all the way and adjust the kickdown linkage so that it will just slip back onto the carb lever stud.
Make sure that the shift linkage is gets all the way to the (1) range. You can (with the engine off) put the trans in the 1 range, then under the car, disconnect the linkage, push the shift lever on the trans as far forward as it will go and see if the linkage drops back into the shift lever. There is also a plastic bushing on the shifter where the linkage connects that could be a problem if it is worn oversize or if the linkage is worn between the shifter, bellcrank and the transmission.
Sorry if some of this stuff sounds oversimplified, with these cars being so old, it's hard to know what the history of the car is, if all the parts are there, if they are stock and so on. If all these things appear to be ok. There could be a band adjustment or low/reverse servo piston issue. Let us know what you find.