Conditional replacement element-wise

I would use MapIndexed[] + KroneckerDelta[] for this:

MapIndexed[# /. jj -> Apply[KroneckerDelta, #2] &, A, {2}]
   {{2 AcD am g1, 1, 3 + g}, {2 g1, g1, AcD}, {0, g1, AcD + am}}

This works by extracting the Diagonal from the matrix first, setting jj in the diagonal to 1 and adding it again to the matrix, where all jj's have been set to zero:

A = {{2 AcD am g1 jj, jj + 1, 3 + g}, {jj + 2 g1, g1 jj, 
   AcD + jj}, {jj g1, g1 + jj, AcD + jj*am}}
setDiagonalEntries[m_?MatrixQ] := 
 With[{d = Diagonal[m] //. jj -> 1, 
   o = (m - DiagonalMatrix[Diagonal[m]]) //. jj -> 0}, 
  DiagonalMatrix[d] + o]
setDiagonalEntries[A] // MatrixForm