The usual idiom is "I don't have to dig".
In speech it is possible to express "I can (not dig)" by intonation, but there is no customary way to manage that in writing. *
Edit in response to the edits to the question: in those cases where you really are talking about the ability not to, as opposed to the lack of obligation: no, you will need to use a perphrasis, probably "able not to", as in your example. There isn't an unambiguous way of using "can" for this meaning.
Slightly off topic: the scope of negation can vary a lot between different languages. English "must not" ("obliged not to") is not a translation of German "muss nicht" ("permitted not to").