First, build your own mailing list - people that you email once in a while about your stuff. For those people, talk to them in your way - whatever that way is. The people that like it will stick, and the people that don't will leave. Don't try to sound like somebody else. Don't try to obey the whims of some fool who says you sound spammy or "too personal" or whatever - in fact, tell those people they would be happier if they unsubscribed.
There are probably some lists out there where you might think it is spammy, but others don't. In time, a crowd forms that likes that sort of thing.
If it sounds spammy by your standards, then reword it.
how do I reach out to people from each of these sectors and ask them to (1) support the crowdfunding campaign by buying the game and (2) support the crowdfunding campaign by recruiting other people from within their spheres of influence to buy the game, all without coming across as a slick-haired salesman or a spammer?
I get contacted by lots of people asking for support of their kickstarter. Sometimes the product has nothing to do with permaculture - so we say "post in blatant advertising or crowdfunding." and then we are done. But if they are regular users or if they worked to help on one of my kickstarters and actually brought in a few buyers? Then I'll do more.
So, my message to you is: build your own collection of things you can do to help people (which, someday, would help you). Then help others. Then when your kickstarter starts, use those tools to help your own kickstarter and then turn to the others to ask for a bit of reciprocation.