What does the exterior look like?

You need to deal with the water issue, first, or it could end up costing you more in the long run. Listen to your husband. Find a respectable, license and bonded contractor with no complaints against him and hire him to get it done.

Your balconies may not have been built correctly. The balcony floors should have a slight fall away from the house to keep this from happening.

How about some closeups?

You need mechanical flashing between the subfloor and the bottom of the threshold that extends up the jambs. The doors have to be pulled, the flashing installed, and the doors re-installed. I've done this job several times; never a callback.

I've got pictures around Houzz somewhere.