The former Kidderminster striker first headed wide in front of an open goal and then blazed an injury time shot over the bar as he looked a sure bet to seal the three points after being put through on goal.
The draw means the Shrews are still looking over the shoulder as their slump continues but at least new boss Simpson can be pleased with the way his side fought back from a goal down.
Carl Baker had given the home side the lead with a superb 32nd minute free kick - his tenth goal of the season but first since January.
Morecambe then came close to extending their advantage early in the second half as Matty Blinkhorn and then Henry McStay smacked volleys against the crossbar with Scott Bevan beaten.
After the reprieve Shrewsbury came back strongly and thought they had levelled on 63 minutes when Dave Hibbert found the net from close range, only to see his effort ruled out for offside.
But, to their credit, Shrewsbury continued to push forward and received their reward 20 minutes from time when Hibbert took advantage of Jim Bentley's misjudgement to cross for Marc Pugh to score from close range.
The game then swung from end to end as both sides pushed for a winner.
Bevan did well to save from Baker and then Paul Lloyd before substitute Constable contrived to miss his two glorious opportunities.