replacement – Replace element within a list of lists with MemberQ condition

I’m trying to replace the second element within a list of ordered pairs depending on whether that ordered pair exists within another set.

shortlist = Flatten[Table[{i, j}, {i, 0, 3}, {j, 1, 2}], 1]
sublist = {{0, 2}, {1, 1}, {2, 1}}

I want to do something like this

shortlist //. x_ /; !MemberQ[sublist, x] -> {x[[1]], 0}

where the result is

{{0,0},{0,2},{1,1},{1,0},{2,1},{3,0},{3,0}}

where the ‘x’ components of the list remained the same, but those that were not members of the sublist had their ‘y’ components changed to zero.