It's right there in the title, Shmoopers. This poem isn't called real song, or my song or even just plain old song. Berryman named this poem (and those 300+ other ones) "Dream Songs" for a reason: he wants to orient us to an altered reality.
And the tranquil hills, & gin, look like a drag and somehow a dog (14-15)
Things start to feel pretty dream-like here in the third stanza. There are unexpected, dramatic jumps from topic to topic, as in a dream. We bet you didn't see that dog coming the first time you read this poem (we know we didn't). Berryman emphasizes the sudden shift and surprise by using enjambment. Line 14 doesn't have anything at the end to slow us down (no comma, no period, no nothin'), so we just fly right through and then "somehow" a dog shows up. Weird.
has taken itself & its tail considerably away into mountains or sea or sky, leaving behind: me, wag. (16-18)
The syntax is pretty strange here, right? Why mention the dog's tail, separate from the rest of the dog? And why isn't the speaker clearer about where the dog goes? All this weirdness and irregularity in the last stanza really reinforce the sense that this poem is taking place in some kind of a dream state. We hope so, anyway. Or next thing we know, pigs will be flying too.