I've taken a full size bike on the Downeaster (paid the $5) and it was no trouble. I've also taken a folder from Boston to Williamsburg multiple times and nobody batted an eye since it was able to fit on the overhead rack. I would think that if you have a folder going on the Downeaster you could just waltz onto the train and throw it up on the rack and no one would complain unless you pointed it out.
FWIW, Woburn in general just sucks for cyclists.