Back-up lights

[Ava66]

Hi All,

Question regarding back-up lights possibly causing a short on my '66. History: one back-up light burnt out. Upon removal and replacement, I noticed socket was terribly corroded & tried to clean it. New back-up light did not work. When I put car into reverse, radio stopped working (turns out that a fuse blew).
So my questions are, How can I most directly fix the problem? Should I replace the entire socket? Should I replace the socket on the other side at the same time considering it is probably corroded as well?

thanks