Osei's first half brace was followed by two goals from Adams after the break, with the Mariners producing one of their most convincing displays of the campaign.
The result kept them two points behind leaders Buxton heading into the final three league fixtures.
Kevin Phillips named an unchanged team for this vital clash, with Sam Wilding retaining his place in midfield as he made his first start at 1st Cloud Arena.
Adams and Jordan Hunter were vital to the victory and they linked up as early as the second minute when Hunter's deep cross from the right was headed wide by his fellow wing-back at the back post.
There was another good chance for the hosts before Stalybridge had time to catch their breath, with Jack Bodenham slipping in Sam Hodgson, whose drive was saved by the legs of visiting goalkeeper Grant Shenton.
Hodgson also headed over from a pinpoint Adams cross as Shields kept up the pressure, and the sense a goal was on its way became stronger when Osei strode into the box and rattled the outside of the post.
The deserved breakthrough arrived on 18 minutes with a fine piece of play as Robert Briggs released Hunter down the right and he raced to the byline before pulling the ball across the face of goal, where Osei was on hand to tuck it into an empty net.
Shields were unfortunate not to double their lead when Briggs presented Hodgson with an open goal and the striker converted, only to see the assistant referee's flag deny him after a marginal offside.
They did not have to wait much longer for goal number two, with Briggs again proving to be the provider as he threaded a through ball to Osei and he rolled it under Shenton to double his tally for the afternoon.
Hodgson did not get on the scoresheet but he was a lively presence throughout his time on the pitch. His latest chance saw him hook a shot over after a flick-on from strike partner Osei.
The Mariners will have felt unfortunate not to go in at half-time with a more comfortable advantage. They had further opportunities in the closing stages of the first half and the closest they came was a curler from Briggs which dipped just past the post.
There were also some sights of goal at the other end, with Conor Falls firing wide from 20 yards, Charley Doyle heading not far wide from a Douglas Nyaupembe cross and Stephen Brogan having an effort saved by Myles Boney.
Shields controlled possession well after the break and had a golden opportunity to make it three in the 49th minute.
Hunter again found space on the right and another superb cross from the right-sider picked out Osei, who was unable to clinch his hat-trick as he headed wide.
A second assist of the afternoon was provided by Hunter soon after, with Adams the beneficiary.
Shields worked the ball from left to right and Hunter beat his defender before delivering another tremendous cross which Adams headed past Shenton to effectively seal the points.
There could have been a third assist for Hunter when he found Hodgson in the next Mariners attack, but he was denied by Shenton.
Osei and Hodgson were then withdrawn as the fresh legs of Cedric Main and JJ Hooper were introduced to the fray.
Main proved to have an instant impact. He broke away down the right, twisted and turned his way past Sam Wedgbury and squared the ball to Adams, who rifled it into the top corner to match Osei's tally for the afternoon and cap an impressive afternoon for the Mariners.
With just three matches remaining, every moment is now crucial in this exciting title race and an Easter weekend double-header could go a long way towards deciding the destination of the championship.
Shields kick-off a brace of fixtures in three days with a trip to FC United of Manchester next Saturday before hosting Whitby Town in their final home league match of the campaign.
Phillips has won all seven of his home matches in charge of the Mariners, with the team itself having won its last nine matches at 1st Cloud Arena.
The former Sunderland striker was pleased with his side's performance against Stalybridge.
He said: "I thought we were very good.
"Some of the patterns of play we've been working on over the last few months really came out in the game and the work-rate of the team was excellent.
"It was potentially a tricky game against a side which on the face of it had little to play for, because they will have wanted to end our good home form.
"I made it clear to the players that we couldn't allow that to happen, and they approached the game so well.
"There was still some slight disappointment because we could have won by a lot more goals and goal difference might come into it at the end of the season, but I can't argue with another four goals at home, a clean sheet and three points."
South Shields: Boney, Hunter (Tee 75), Adams (c), Shaw, Morse, Bodenham, Wilding, Briggs, Hodgson (Hooper 72), McGowan, Osei (Main 72). Subs not used: Rose, Jenkins.
Goals: Osei (18, 28), Adams (61, 73).