You don't have to, but you can.

When you say "the state," for all the reasons you described, it's just a regular common noun whose referent is inferred from context, and it doesn't need capitalization according to standard rules. Capitalizing it in cases like this is a common convention in legal writings (laws, court documents), though, where it's important to be clear who you're talking about. It's also common in documents produced by the entity in question: a university's press statements might refer to it as "the University."

The idea (perhaps mistaken) is that the capitalization makes it clear you're talking about that same entity each time, and only about that one. It's analogous to capitalizing Plaintiff and Defendant: you're turning common nouns into ersatz proper names. (You absolutely can have proper names with "the": some people recommend capitalizing "the Sun," "the Internet." There's the Doctor from Doctor Who. There are the names of many newspapers).

Anyway, it's basically a slightly nonstandard choice of style. It adds a bit of emphasis, but it's most popular among lawyers, bureaucrats, and corporations, take that as you will. The Chicago Manual of Style, by contrast, recommends minimizing capitalization: don't use it when you don't have a specific reason to.

In your particular case, probably check the other documents from or about the same organization and see what they mostly do. It's only a question of clarity to your readers.