Granted, I lived in a primarily dry climate, but I also have used many Sentry safes. Always put my coins in zip-lock bags inside of the safe. 99% of the coins were fine. For the majority of their life (in my possession), in zip-lock bags in a crawl space.
Let's put it this way - don't. Coins are best stored in a cool, dark environment. Being warm promotes toning. The best place to store your coins (or safe containing them) is an interior closet that has no walls adjoining the exterior of the home. In other words, all 4 walls of the closet, if you went through them, you would not be outside. This is because an outside wall is subject to the temperature outside and since outside temperature fluctuates, so would the inside of the closet. What you want is for the temperature to remain as constant as possible in the coin storage area.
Unless the base metal coins were selected for quality your loss might be a lot lower than you're thinking. Sure sentimental value is probably destroyed but the coins can be spent and the proceeds will cover a large part of replacement costs... ...especially mint sets. I don't believe there's anything that can be done about carbon spots but if there's anything special in there you can get them conserved or try it yourself. Don't spend things just because they have haze since this can usually be removed. If you replace the modern sets you can even hand select sets and come out ahead.
The base metal coins are ruined beyond repair. The silver and gold will be fine. Change all the holders. Remove the silver and gold and place in new holders. Coins like the modern proof sets, the 1983 half dollar for example, spend them and buy new ones. You should be able to buy a nice 1983 proof set for around $4. All the packaging that is moldy? Destroy it and replace it with clean ones. Packaging for most modern coins sets can be found with a little searching for a modest price. I have tons of it here at my shop. If the silver and gold coins have any residue or mold on them, it can be removed with a double or triple bath of alcohol or acetone. In 1996, my town flooded and many had coins in bank vaults that went under water. Coins had silt, mud, worms and fish (no kidding) all over them. I helped many recover what they could, which was quite a bit so don't despair, much can be salvaged with a little effort. Oh yes, and find a new way to store your coins!
WARNING DO NOT store delicate items directly in your product SentrySafe products which offer fire protection have a patented insulation that has a high moisture content. If you choose to store delicate items such as jewelry with working parts, watches, stamps, or photos in your product, we recommend putting them in an air-tight container, prior to placing them in the product for storage.