We just bought a raised ranch in a RI with an outdoor shower. The lower level of the house has seemed damped, and a dehumidifier has been helping, but this weekend we noticed the carpet in one closet was VERY damp - and it was right next to the wall with the outdoor shower, above! We are thinking the shower drainage is poor, and/or there is a problem with the foundation.

What needs to be done, and who should we call?

If you have a well keep all water off and see if the pressure gage drops and the well kicks on. If its township water off and see if the meter still slowly turns. If so then a small leak. This will help taking that out of the picture.
To do a slow test for the drainage can you not use that shower and see if that changes things. Of course if the house drainage is also hooked in that line this will not be a good test.

The person to call is a plumber. Do the pressure/usage test so when the plumber gets there he has more to go on.

Can you give better details on the shower and the condition its in or a Picture would be much better. It cold be as simple as the seal around the drain is bad. How often is this shower used?

Get back so everyone here has more to go on..

There should be stop and waste valves for the shower some where inside the house. Turn these off and wait to see if dampness goes away. Will a few days.
Sounds like someone either forgot to turn off during winter months or turned off but did not properly drain pipes. Now you have a busted pipe due to freezing.