Here’s what The Chicago Manual of Style 8.55 has to say about use of initial capitals in references to political divisions:
Words denoting political divisions—from empire, republic, and state down to ward and precinct—are capitalized when they follow a name and are used as an accepted part of the name. When preceding the name, such terms are usually capitalized in names of countries but lowercased in entities below the national level.
CMOS gives as an example “Washington State; the state of Washington.”
In my experience, 99.6 drafters out of 100 would say the State of Washington. It’s generally best to follow standard English usage in such matters, so I’m inclined to join the 0.4 for purposes of contract references to, for example, a certificate of good standing from the secretary of state of the state of Delaware. Any thoughts?
Oh, the lowercase letters in secretary of state? See CMOS 8.21.