Salinity will absolutely go up by replacing evaporated water (zero salinity) with saltwater. The salt left behind by evaporation will still be in the tank, so salinity will go up proportionally to the total amount of salt in added saltwater, even if the salinity of the added water is the same as what you intend your tank salinity to be. The evaporation process only removes water and leaves salt behind, while adding saltwater adds salt and water to replace the evaporated water (only).
Think about water (W) and salt (S) as variables.
Adding initial saltwater is W + S, evaporation is -W, adding RO/DI or tap water is +W, adding saltwater is +W + S.
So, the equation for evaporation replaced with RO/DI water is:
W + S - W + W = W + S (original salinity)
For adding saltwater for evaporation replacement, the equation is:
W + S - W + W + S = W + 2S (raised salinity), W + 2S - W + W + S = W + 3S (raised salinity even higher), and so on for each time you replace evaporation with saltwater.
Edit: Of course, the increase in salinity would not be double, triple, as could be taken from my equations above, since the added salt would be proportional to the percentage of total volume in the water volume added for top-off. Salinity would actually climb slowly, but eventually would have detrimental effects.
This is one of the most basic principles to understand for keeping a saltwater tank and should have been understood before lives are at risk. I am guessing this was your comment that made
@brandon429 concerned with your general level of marine aquarium knowledge.