I hope the electrical service and electrical panel in your home is not the original installed in 1962.
As for your safety concerns using the ungrounded outlet your audio system is plugged into, you could buy one of these to feed the power to your audio system for now.
Example of:
More examples to choose from.
https://www.homedepot.com/b/Electrical-Electrical-Cords-Extension-Cords-GFCI-Plugs/N-5yc1vZcger
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