Everything you need to process the metal is a short walk from where you are. Get a few surtling cubes, build a forge, and process the metal there.

I usually look for a stone tower that has several copper rocks around it and setup shop there.

Digging underneath doesn't seem like it would be worth the effort either, but I've never tried it.

Did you make a bronze pickaxe and axe? If not, you should. It makes the process go much faster.