A "calorimeter" might be used. This measures the temperature rise of a secondary fluid when indirectly exposed to a sample of the exhaust steam.
Another method is to use a "tracer" - usually some radioisotope - that has a much greater solubility in liquid water than it has in dry steam. (Common salt is one example.) The amount of salt in a sample of exhaust steam is then correlated to the exhaust moisture.
Sampling techniques become important with any of these methods; you must insure that your original sample is representative of the bulk quality (i.e. no additional moisture in certain locations of the exhaust trunk because of flow irregularities.
The tracer method was pioneered by GE - Large Steam Turbine Division (Schenectady, NY). Various of the principal engineers from those facilities have been and/or are still on related ASME Committees. I'm sure that some have started consulting companies of their own.
I recommend a web search.
If you're also considering purchasing test services, the credentials of the consultants would ideally include testing experience with one of the manufacturers of large steam turbines (GE, Westinghouse, ABB (or its predecessors), Toshiba, MHI, etc.)
And, by the way, I think that you should consider purchasing test services, at least for the first time.
