What medical problems are likely, after seasickness?

Constipation.  Sunburn.  Minor bruises.  

Constipation is extremely common, and may be the subject of crew discussion.  Lack of lower body exercise combined with a low-fiber diet, possible dehydration and a disrupted sleep schedule is a formula for serious constipation. Multi-grain hot cereals are usually effective, but everyone has their favorite remedy and should bring what works for them.

The tropical sun, casualness about sunscreen, and reflection off the water can combine to give you a dose of UV that is greater than you really intended. Slop on the sunscreen.

Especially early in the trip you may get tossed around, coming into contact with hard things.  Usually, this results in minor bruising or strained muscles.  In more severe cases, a cracked or broken rib may result.  Install and USE handholds.