Should "State" be capitalized on its own?

I would say not, because in the second sentence it is no longer specified, but then I'm a mere Brit.

Would you write Maine state, New Hampshire state, or even Rhode Island and Providence Plantations state or would you have the word state camelized? FYI, the full name of Rhode Island is the longest name of any state in the Union.

See, if you cannot write Maine State (or Rhode Island and Providence Plantations State), rather than Maine state, with a clear conscience, than for equality's sake you should not be able to write New York State or Washington State.

Would you write Pennsylvania State or state?

When you write Maine State Lottery or New Your State Police, you would not actually be writing Maine State or New York State, but Maine {State Lottery} and New York {State Police}, especially that these are proper names of entities within those states.

However, due to state/regional patriotism, people in Florida for example, might actually write Florida State. That would be like asking if the word gods should be camelized. Therefore, people outside the patriotic sense of Florida should write Florida state, as people with no allegiance or faith in gods would write gods, uncamelized.

However, New York is a special case. There exists a need to differentiate the city of New York from the state of New York. Since we would write as New York City, normalization of naming convention would require us to write as New York State, regardless of where your allegiance lie.

I think the simplest way to say this is "If it's part of a pronoun it's capitalized, otherwise it isn't".

Eg "Washington State students are protesting on tuesday"

"Students are protesting against the state government on tuesday"

"Washington State is one of the more northerly states of the USA".

Almost five years and 82,000 views after this question was posted, no one has yet cited the advice from an actual style guide. Let me break the ice by quoting advice from four major U.S. style guides.

From The Associated Press Style Book (2007):

state Lowercase in all state of constructions: the state of Maine, the states of Maine and Vermont.

...

Do not capitalize state when used simply as an adjective to specify a level of jurisdiction: state Rep. William Smith, the state Transportation Department funds.

Apply the same principle to phrases such as the city of Chicago, the town of Auburn, etc.

AP's position on lowercasing state also applies to instances where the word follows the state's name:

state names Follow these guidelines: ... Use New York state when necessary to distinguish the state from New York City. Use state of Washington or Washington state when necessary to distinguish the state from the District of Columbia. (Washington State is the name of a university in the state of Washington.)

From The Chicago Manual of Style, sixteenth edition (2010):

8.50 Political divisions—capitalization. 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 [cross reference omitted]. Used alone they are usually lowercased.

[Relevant examples:] Washington State; the state of Washington[;] the New England states

From Words into Type, third edition (1974):

Divisions of the World or of a country. Cap names of the divisions of the world or of a country.

[Relevant examples:] North Atlantic States[;] New York State

Lowercase: state {used in a general sense and when it does not follow a proper name: state of New York, state of Oklahoma} ...

From Allan Siegal & William Connolly, The New York Times Manual of Style and Usage, revised edition (1999):

state. Capitalize New York State, Washington State and formal references to any state of government: The State of Ohio brought the suit. Lowercase state in references to a geographic area (They drove through the state of Illinois) and when it stands alone (The state sued the city). Capitalize when State appears with the name of an official agency or with an official title that is capitalized: the State Education Department; State Treasurer Pat Y. Berenich. Use state in reference to New York and Washington when necessary to distinguish them from the cities, but omit State if the context is unmistakable: The governors of California and New York have similar powers. Nebraska's population is smaller than Washington's. Lowercase in the general sense: affairs of state.

Overall, the four style guides I've cited offer fairly consistent advice on several points, although AP breaks with the other three on the appropriateness of initial-capping State following New York or Washington (in part to make clear that the state is intended and not New York City or Washington, D.C.) and doesn't make an exception to the general rule against capitalizing state in "state of X" constructions in cases involving formal references to state governnments of the form "State of Idaho." All four guides advise against initial-capping state when it appears in informal, descriptive contexts.

With regard to the poster's question, I think, Chicago, Words into Type, and the New York Times would endorse this approach to capping/lowercasing the posted text:

Higher-education spending, clout, and influence in New York State are substantial. Within the state’s borders...

But AP would favor this treatment:

Higher-education spending, clout, and influence in New York state are substantial. Within the state’s borders...

The most surprising split in preference is the one between AP and the New York Times. On most points, the Times follows AP style—which makes sense because AP style governs the form used in AP wire service stories. The Times's decision in favor of "New York State" and "Washington State" means that someone at the newspaper has to go through all AP articles and initial-cap the word state whenever one of those two phrases appears.