After the imminent nuptials of Prince Harry and Meghan Markle, the couple will be given the royal titles of...wait for it...Duke and Duchess of Sussex! Smashing, we know. Sussex was the odds-on favorite due to the fact that, well, there hasn't been anyone occupying that spot for a while. Harry (whose real name is Henry, who knew?) also holds the titles of Earl of Dumbarton and Baron Kilkeel, per Vanity Fair.
At their 2011 wedding, Prince William and Kate Middleton became the Duke and Duchess of Cambridge, though many in the public and the media still refer to them as some variation of "Prince William and Kate Middleton." Similarly, we expect that the royal couple 2.0 will be known to the world as Prince Harry and Meghan Markle. "Her Royal Highness The Duchess Of Sussex" may sound like an unwieldy thing to sign on your checks, but most likely the royal couple are not actually signing checks at all.
In case you're interested, here's a bunch of specific information about this Dukedom that you can whip out at parties, if you go to really weird esoteric parties where people talk about British history:
To get to your real question, "she's a princess now, right?" the answer is...no. Except, she is a little bit. Princess, as a title, is reserved for women born into the royal family, like Princess Charlotte or Princess Anne. Those who marry in have other titles, like Duchess. But on the other hand, it's not totally wrong to refer to the wife of a prince as a princess. We, the collective English-speaking people of the world, get it. Princess Di was Princess Di, you know? According to Vanity Fair, "Markle also will be known as Her Royal Highness, Princess Henry of Wales, corresponding with Harry’s standing as Prince Henry of Wales."