I would first suggest, if you have a say in the matter, try to cover songs that you can connect with on a personal level (that you don't find to be nonsensical). I'm sure you chose this one for some reason, maybe it's just about the instrumental aspect, but you will find memorization easier if you can find personal meaning in it.
The lyrics aren't particularly obvious, but I wouldn't say they're not poetic. There are some line breaks I might add to the source you provided that might make it clearer, but I would say it has a very metered feel that flows well and is also highly rich in metaphor. Check out some analyses of poems by e.e. cummings to perhaps broaden your palate for poetic metaphors. :-) In other words, I think there is definitely meaning in this poem; (there almost certainly is to Mr. Johnson) and if you can find it, it'll only make your whole experience with the piece more enjoyable.
I read the poem before hearing the music, and I'll bet part of what's tripping you up is that the lyrics as sung have a very constant rhythm and not much melodic contour. I would suggest extracting the lyrics from the song and treating it like poetry. Live with it like that for a while; see if you can read it differently from the rhythm you know in the song.
In this format, you can also chunk the poem down into component parts; similar to how one would analyze a song in, say, AABA format. See if you can uncover the logic of why and when he repeats lyric material, why the choruses happen when they do, how lyric material relates to music material.
One additional straight memorization tactic you can try is simply writing the poem down, word for word, exactly down to the letter from memory. Find a bunch of looseleaf, error check yourself each time until you've got it perfect, and then rewrite it a few more times for solidarity.