Has anyone traveled to Turkey with I-797A as a supporting document?
"Form I-797A: Is issued ONLY to people who are currently in the U.S. and are undergoing a change of status (F-1-H-1, H-1-H-1, l-1-H-1, etc.).
The “A” is for change of status approved, which means the applicant can continue to remain in the U.S. and work. Hence, I-797A will have an I-94 attached to it to endorse the status approved."
Moreover you have already mentioned that you have I-797A (H-1B visa extension document) valid till July 2019, means you are a legal resident and can continue to work and remain in the USA till 2019.
As you have already visited the most trusted authoritative place (Turkish consulate in Boston), therefore you will have to follow their instructions means simply apply for Turkish e visa.
Here is the website where you can simply apply online for evisa:
Turkey e visa
You must meet all the requirements listed below in order to obtain an e-Visa and
I-797A (H-1B visa extension document) is a valid supporting document.