How do I write LGBT characters without looking like I'm trying to be politically correct?
##Don't Try. Do.
Actually BE "politically correct." "Politically Correct" is originally (and in my view still is) a pejorative and dismissive term for being fair and unbiased in your writing. Anybody that accuses you of "trying to be politically correct" is insulting you for trying to write a fair and unbiased portrayal, so ... screw them! The only way to appeal to them is to never violate their unfair and biased cultural expectations. If that is your market, then go for it, but guess what? They are a minority and niche market in the overall book / writing market.
To be fair and unbiased, avoid using stereotypes, they don't apply. There are both butch and girly lesbians, there are both manly and camp (effeminate) homosexual men.
Avoid positive stereotypes as well: Don't idolize them, don't make them paragons, give them flaws and shortcomings like any other character. Don't make them exclusively villains.
Don't hide their sexuality in public anymore than you would a heterosexual couple, and don't try the cop out that "some heterosexual couples are just not very romantically expressive in public, so I chose that as a trait for my homosexual couple."
Don't assume because they are LGBT they are promiscuous or into group sex, either. They most probably are not.
##Make them average.
In real life, LGBT or asexual persons really are minorities; a super-majority of people, by nature or culture, truly believe they are heterosexual.
A minority character (relative to other characters in the work) presents a conundrum for the writer, because there are not enough of them to show any spectrum of traits for that race, religion, sexual orientation, size (say of people with dwarfism), or person with a disability.
Because you cannot show such a spectrum, your character becomes the designated representative of that minority. It is fairly easy on the Internet to find details of what that particular minority finds offensive, what an unbiased portrayal looks like from their perspective.
So just BE fair and unbiased and make your character unoffensive to the group it represents. Make them the average. If they have exceptional traits (as many characters must), try to pick traits that don't have anything to do with whatever that minority objects to as stereotyping or insulting.
You can give yourself leeway on "being average" if you have multiple instances, the more you have, the broader a spectrum of traits you can portray, but I would still try to vary them in their non-sexual traits instead of sexualized traits (like promiscuity or romantic aggressiveness).
The average sexual trait for all humans is ... we like sex with mutual sexual attraction, in private and alone with our partner. We like romance, love and closeness, and we are demonstrative without being over-the-top in public. Remember Al Gore's ridiculous passionate kiss of his wife on TV? That's heterosexual but was widely ridiculed, as if he was trying to prove something, because it was beyond the norm of public expression. We hold hands in public, we kiss on closed lips (unless we are teenagers), we embrace without grabbing any sexualized body part.