नहीं is always placed adjacent to verbs.
With regular verbs, it can be placed on either side (with नहीं being after the verb when you want to emphasise the negation). With auxiliary verbs like है, नहीं is always placed before the verb.
है is optional in negative sentences only when there is another verb in the sentence.
In this case, है is the only verb (दुखी is an adjective) and cannot be dropped.