Yes, your suspicion is correct. However, there are foundations and other sources who like to support government entities with grants or other means. For example, a corporation with headquarters in Springfield might make a grant to build a recreation center there. Often, the recreation center or the school or the library might create a "friends of" group which is the 501 c 3 nonprofit fundraising arm of the government entity.
That said, I am not that familiar with private or corporate funding for state entities. I am familiar with many partnerships (e.g., the state gives x to build a park and a foundation gives y), but not direct grant making (e.g., foundation gives y to the state to build a park).