Ours were kind of rough when we bought Adagio, and it took some experimenting before I found a good option. I say 'good' but it's still a.fair amount of work.
Folex brand stain and mildew remover was the first assault. Spray it on heavily, let it soak for several (10-15) minutes then scrub with a bristle brush.
Next, I use a "Green Machine" miniature carpet cleaner with a mix of pet stain/odor removing carpet shampoo and hot water. Using this, I soak, scrub and wet-vac until it looks pretty good.
Finally, I again used the Green Machine with water and about 3%-5% bleach and it really made it look nice.
I did only a couple of cabinets at a time so I could use portable fans to make sure they dried out nice and quickly.
I was worried that all this chemical attacking would possibly loosen the adhesive, but so far so good after about 5 years.
Generally, we leave cabinet doors open if it's reasonably do-able just to keep some airflow going.
It's a tough scenario with such a tight boat. Good problem to have, actually :).
SM 209, Adagio