In my opinion, it would not be a string issue. The variable cap has stops at the extremes. If string was slipping, the needle would stop at a point before the bottom (or top) of the scale.
That does not sound like the case as you described. A more likely situation is that some tuning components are either failing or their values drifted. These values affects the alignment and bandwidth range. The service manual will have instructions on how to adjust the high and low range. This will involve turning some of the transformer coils or varicaps but you'll need a rf signal generator to do this. Typically you'll use the rf generator set to a specified low range frequency (i.e. 88mhz) and make adjustments until the tuner receives the signal outputted by the generator. Then you'll tune to the high end (i.e. 108mhz) and set generator to that frequency, and tuner again adjusted to best reception. This is repeated several times. After adjustment the entire bandwidth tracking should be quite accurate.
However this adjustment is a waste of time if the components are not stable and their values continue to drift. If the local oscillator is not set to proper frequency, then the received frequency won't be correct either.
A lot of people think adjusting the needle position is a fix, but in reality, it only ensures that the needle is correct at that particular frequency because tuning to a different location of dial will result in the needle once again in wrong position. This is because tracking will always be off if tuner scale and actual receiver bandwidth is mismatched.
So in summary, you can fix this if you follow the precise steps prescribed in the service manual, you fix the causes of drifted component values or local oscillator frequency, and you have or have access to some pricey lab equipment.
