Each line in the first two verses can stand by itself, but in the third verse, the meanings seem to extend beyond the line breaks. Poetically and musically, it's structured as:
Woke up this morning; I was feelin' quite weirdHad flies in my beard; my toothpaste was smearedOver my window; they'd written my nameSaid, "So long, we'll see you again"
but, as I understand it at least, some of the semantic breaks don't follow the line breaks:
My toothpaste was smeared over my windowThey'd written my name, said, "So long, we'll see you again"
"My toothpaste was smeared" seems incomplete without "over my window" to specify where it was smeared. That the line "Over my window they'd written my name" would otherwise be inverted seems to confirm my reading.
This sort of discrepancy between the structure and the meaning, with the line breaks falling in strange places, matches the narrator's "feelin' quite weird."
Additionally, it occurred to me that there's some resemblance between "Mr. Spaceman" and Dylan's "Mr. Tambourine Man." The titles are obviously similar, and the chorus of each has lines that start with "Hey, Mr. Spaceman/Tambourine Man." Although the two songs are in different keys, the entirety of "Mr. Tambourine Man" and the verses of "Mr. Spaceman" have the same chords: G major, A major, and D major, sometimes even appearing in the same order and for only one measure each.