This is my opinion, so take it for what it's worth.
Water stains are totally different than water damaged cabinets, structurally speaking.
Ask yourself if you want close to perfect results, or just something you can live with. I have dealt with this type of damage previously, and I would not do what I did to repair them, way to many hours of labor involved.
From what I can see is the lower left corner on the left side, and back bottom right corner are an issue. What is the location of the damaged area shown in the first photo posted? Trying to repair to near perfect condition will be tedious and time consuming, even if possible. Put it this way, if you build a new cabinet and place new veneer on both cabinets, you will save many hours of labor and your results will be perfect cabinets!