"A-" before a verb was a prefix quite common in 16th C. English. It is still, today, quite common in Appalachian English, in the US, which is where Dylan no doubt took his influence.
It can mean "engaged in", as in "He's a-runnin! And fast!", or "She's a-birth, and there's no point in hoping she'll not."
It can also mean "motion to, into", as in "I'm going a-long", "I'm going a-bout", "I'm going a-round", "I'm going a-breast".
It is not today used in formal English, but it is an archaic usage from the 16th C. English (as one sees in the words "around", "about", and "abreast"), so it is not an "Americanism". It is an archaic form of English that survives in America (and with consideration for the powerful influence this prefix worked over our prepositions, I'd suppose it survives in parts of Britain, as well -- Scotland, perhaps?).
Being a colloquialism, its usage is largely regional, and so hasn't gotten enough attention to register on my personal "research radar" -- I, having lived in/come from Appalachia, find it rather intuitive. But my linguistic skills aren't sharp enough to describe precisely how the usage might work, unfortunately.
This is the sort of thing that, if you want to know how it's used, you've got to move to the place where it's spoken and hear it in speech for yourself.