In propagating errors you add variances, not standard deviations.
You would square both standard deviations, and then add them. Then square root to get the standard deviation of the system. In which case your error, if assuming the same standard deviation on measurements before and after becomes Std Dev * root 2.
The problem can be broken down into
Difference + Error(overall) = (Before + error(Measure) - (After + error(measure)
but my heads to frazzled today to work it further than that