"I should go now."
Translation:अब मुझे जाना चाहिए।
Your are right. You can say मुझे अब जाना चाहिए। But the emphasis sounds different to me. The given answer just means that your are now saying you have to go. Putting अब next to जाना sounds to me like you are saying you have to go now (immediately, emphasis on "now").
Chahiye (as per gender).... is used both in the context of "wanting" as well as in the context of "should". Generally when it is used in the context of latter.... there is another transitive verb in the sentence, in this case "Jana" .... and therefore chahiye becomes an adverb and means should.