Where is this problem coming from?

You've got a short to ground somewhere. You mentioned installing a new alternator. Make sure the big stud on the back where the main wire is connected isn't touching anything.

Inspect the harness starting at the fusible link and trace the wires. You could disconnect the bulkhead connectors and try a new link to rule out the dash harness.

The fusible link is doing it's job. It should be like a 16 GA link, not a fuse. And the auto parts store ones are fine.