Trademark vs Not

If I understand right, non-profits have to be trademarked, correct? I've done the search, and did not find one with our chosen name, but I just found a for-profit company in a different state with the same name. Is that a problem? Does one of us need to change the name?

Also, does arrangement count? We're planning to incorporate and trademark as one word, and they are using it as two.

TIA