Can I begin a sentence with "and?"

Yes, you can. But first, let’s look at why some might advise against this. “And” is a coordinating conjunction. It links words, phrases or clauses:

That morning, I woke up with the intention to feed the chickens and find a new farmhand.

He has the respect of the players and he knows how to leverage it.

Coordinating conjunctions usually join elements within a sentence. Some argue that this is their only use and, as a result, starting a sentence this way renders it a fragment. Still, some writers begin with a coordinating conjunction for stylistic effect. Here’s an example from “A Very Old Man with Enormous Wings,” a short story by Gabriel Garcia Marquez:

That was one of the few times they became alarmed, for they thought he was going to die and not even the wise neighbor woman had been able to tell them what to do with dead angels.

And yet he not only survived his worst winter, but seemed improved with the first sunny days.

People often begin with coordinating conjunctions in spoken language. So, your dialogue might reflect that, too. You can see this in the short story “What We Talk About When We Talk About Love” by Raymond Carver:

“Terri and I have been together five years, been married four. And the terrible thing, the terrible thing is, but the good thing too, the saving grace, you might say, is that if something happened to one of us—excuse me for saying this—but if something happened to one of us tomorrow I think the other one, the other person, would grieve for awhile, you know, but then the surviving party would go out and love again, have some else soon enough.”

As with all stylistic choices, be thoughtful in your usage.