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It sounds to me like a router issue.

Is the old computer still connected to the same router? If yes, then your new computer cannot use the same connection or listening port. Routers seem to reserve them for each computer & each program. eg: if LW on the old computer is using port 6346, then LW on the new one should use a different port such as 64049.

There is a chance you may need to port forward your router for the LW on your new computer. That's why I am asking for extra information above.