In your example, a-swing means simply swinging.
OED defines this use of the prefix a- as:
- With nouns and verb stems, forming adverbs (and derived adjectives and prepositions) expressing activity, position, condition, etc. Now chiefly poet.
Beneath this entry, tiny type tells us:
A selection of rarely occurring formations is given below.
The two most contemporary of these are:
1989 T. Tryon Night of Moonbow iv. iv. 261 Harpo took up a position outside the window, tongue still hanging a-pant, earnestly cocking his head.
1999 Esquire Mar.33/3 Cynics often wonder, aloud, and asmirk.
While the usage in your example may serve mainly to satisfy the demands of meter in verse, it is from the use of this prefix (according to Jesperson, at least) that, over time, such words as aswim, agape, afloat (which dates to 1023 CE!) and awash have evolved. Jesperson goes so far as to claim that they may
"...for all intents and purposes be regarded as a new type of present participles, but their use is often more literary than colloquial."
N.b.: Words such as asleep and around have evolved differently; in these, the a- prefix represents the OE on or an with the meaning "on" or "onto", and is prefixed to a substantive such as sleep or round, not to a verb form.