Without a victory since January 19, Shrewsbury have now gone 13 matches without a win as they failed to score for the third successive home match. All of which is in stark contrast to ever-improving Bury, who put more distance between themselves and the lower reaches of League Two.
This hard-fought and often entertaining encounter was settled when Adams struck with just under 20 minutes remaining to secure Bury's seventh away win of the campaign.
With heavy rain falling for most of the afternoon, Bury started with the confidence brought about by their recent upturn in fortunes.
Top scorer Andy Bishop spurned the best of the Shakers' first half-opportunities, although at the other end Bury had a scare when an inswinging corner from Kevin McIntyre flashed right across the face of goal.
After the break Town managed a greater share of possession, but chances remained at a premium with Bury keeper Darren Randolph rarely tested.
The decisive moment arrived on 71 minutes, when Adams cut in from the left and his intended cross sailed over the head of Shrews goalkeeper Glyn Garner and into the net.
The home side created an immediate opportunity to equalise, but central defender Graeme Lee blasted wide from 12 yards and there was more anxiety for the Bury defence when a near-post cross from Shrews' substitute Chris Humphrey was scrambled away for a corner.
Striker Dave Hibbert then headed narrowly wide, but Bury were not be denied all three points and almost extended their lead with a Richie Baker volley that Garner pushed away.