New car stereos usually have two power supplies -
The switched supply for the main sound functions etc
A permanent supply for the memory functions.
If your existing single supply is switched, then you will need to add another fused permanent supply for the memory. You can take this direct from the battery or from the fuse box.
Presuming back in '64, car radios used mechanical memory; manually dial or tune into a station then push one of several numbered buttons to memorize the dialed position. Tune into other stations and push other buttons to memorize several stations instead having to repeatedly tune into a favorite station(s). 100% electronic radios use two power wires, one is ...
Since it's only happening in 1st gear it could be something to do with engine mounts or the rotating of the engine is hitting something. 1st gear puts the most stress on the engine mounts and drive axles.