I went to the museum today with my mug. Good news to report. The mug is real! She agreed with me with the time period. It's around 1840. Claire, (the curator), asked if she could keep the mug for 3 months to do further research. She made connections before I came. One was "The Mercer Museum, which is amazing, I've been there.
http://www.mercermuseum.org/ There was somewhere else, but I can't remember. Claire she had many other places she wanted to check. As far as an appraisal, she could not give me one. They don't have one at the museum. However she did give me three places to contact for appraisers that will really research and then give you the value. Haym Solomon is so rare they only have 3 or 4 items at the museum. He was George Washington's financier. His picture was never taken, but they have his signature. Claire said there is a huge upswing in interest of "Haym Solomon". They just built a statue of him in Chicago. Claire told me she would keep me up to date. What do you think the mug is worth now? Just curious as to your thoughts.