pronouns - Is there a non-transphobic alternative to "he or she"?

For instance, "A politician must be able to think quickly on the spot. He or she must also have no qualms about lying."

I know some people who use "they", but as that both sounds and is ungrammatical, I'm wondering if there is any other concise and non-awkward alternative.

Edit:

Apparently many do consider "they" as correct. This surprised me, because I know someone who actually wrote a grammar book, who told me in no uncertain terms that "they" could not be used (in fact, I'd asked them - ha! - this question before I posted it here). And a quick look online found legitimate sources advocating both positions. I suppose it's one of those issues, such as whether the previous sentence was incorrect for beginning with a conjunction, where it really just depends who you ask. Regardless, to me personally, "they" in the above example sounds clunky at best and incorrect at worst. So I suppose my question can be rephrased as, 'is there any alternative to 'he or she', other than 'they'?"

Secondly, to whoever tagged this as "politically-correct": acknowledging the existence of genderqueer or transgender people isn't a political issue at all, until those made uncomfortable by people unlike them make it one. Trying to avoid excluding and thus unconsciously discriminating against a group of people in your writing isn't born out of a need to be 'politically correct' - it's born out of basic human decency.

Finally, yes, in the example above, I could use a comma and avoid the need for a pronoun at all - but the question I'm asking applies to instances where one may not want to do so (i.e. the sentences are long and stringy enough as it is).