From what I read, if you can get a no objection statement waiver from the sending country, you can bypass the J-1 status, correct? If there is financial obligation, is it possible to settle the financial obligation with the sending government to obtain a no objection waiver?

Is there any way to determine whether the home country's government will issue a "no objection" letter to the US State Department?

Under No Objection Statement, it says:

The home residency requirement for a J-1 holder may be waived if the alien's home country's government can issue a "no objection" letter to the U.S. Department of State indicating that it does not object to the J-1 waiver grant.
How do I get the home country's government to issue a no objection statement? Do I contact the embassy in Washington DC?