After linking verbs we use adjectives (be, sound, seem...).
You even know the term predicative adjective. Those predicative adjectives are connected through copula (linking verbs).
You can find the rules about it easily on the internet.
The verb "feel" can be a linking verb, as it is in your sentence. If the modifier that follows it describes the subject, the modifier should be an adjective -- in this case, "bad". However, you will see and hear "I feel badly" often.
Although that sentence is grammatically sound, I find it unnatural. I think we'd be far more likely to say something like ' I have a hunting dog who has a very poor sense of smell'.
Although that sentence is grammatically sound, I find it unnatural. I think we'd be far more likely to say something like ' I have a hunting dog who has a very poor sense of smell'.