Is the only way to replace the socket for a turn signal lamp to cut the wires and splice the new one in?

 

Are you talking about the ones in the lower part of the bumper?

 

It is possible to remove the pin sockets from the connector and re-install in a replacement connector body. You can pull wires from a donor connector to replaced damaged spades in your connector. A special tool is required. See following video.

Search w/ google for "How to Fix Your Wire Harness Plug Connector on Honda Odyssey".

good luck