NY versus WY or NV - is it really worth it (indemnification)?
First, you noted that you will be running two "distinct businesses" under one entity. Generally, it is not recommended to run two distinct businesses under one entity. It complicates accounting, creates additional liabilities for the 'other' business, will prevent you from selling one of the businesses or even allowing a potential buyer to do meaningful due diligence without first untangling one from the other at a high cost.
Second, there is no advantage for a LLC, that is physically present in New York, to form itself in a different state. New York will still require you to register as a foreign entity and you will end up paying higher fees as a foreign entity. There are some advantages for corporations to form in other states and I recommend that you first consult a business attorney in the event this will be a corporation.
Third, since you are concerned about customers suing you, by being a Wyoming/Nevada company, keep in mind that a customer can file a lawsuit in Wyoming/Nevada and you will have to defend the lawsuit locally there. Are you ready to travel extensively to Wyoming in such event?
If you need help in the matter feel free to contact my office at your earliest convenience.