Side Slip,
I fly an even older 172, a 1958.
I have experienced the same problem you describe but to a lesser degree. There is an AD on the fuel system that requires you to use either left or right tanks when at, or above 5000ft. I did some research some time ago and seem to remember that there had been some vapour lock problems when flying above 5000ft when 'Both' tanks were selected. Apparently using either left or right tanks solved the problem. The fuel selector in my aircraft has a label instructing you follow the AD.
In normal usage I have also noticed it will draw more fuel from one tank or the other. For some time I did not realise that depending on the crosswind strength, the old girl seemed to track better with the upwind wing down a bit, and I just flew her that way! (I know crosswinds are not supposed to effect the aircraft in flight, but try telling my old Cessna that. If I hold the wings level she drifts off with the wind. It doesn't seem to matter if the crosswind is from the left or right. Using rudder to hold her just creates sideslip! I just put down to the old girl being over 50!) Of course it used more fuel from the downwind wing tank because it was a little bit higher. I just keep an eye on the interior mechanical gauges and if the imbalance alarms me, select the other tank for a while.....
Alway use both for takeoff and landing though.
Have fun with your old 'straight tail'.
Regards,
Ian.