How do you decline a draw offer?

In OTB, Do we say “No?” Do we shake our heads? Do we just make a move? I never know how to decline the offer. In the Netflix series “The Queen’s Gambit,” she slowly shakes her head.

Is there something similar to resigning by offering a handshake?

What a weird question. How do you usually say no in everyday life? The most cultured way is "no, thank you" I suppose, but you can pretty much shake your head, ignore it, you can make chicken noises, really anything that clearly shows the game will continue.

In OTB chess, the convention is to make a move and then offer a draw verbally to your opponent.  That way, your opponent can decline through either gesture/verbally or by making a move.

As for resigning, the higher the skill level you go, the more accepted offering a handshake is a resignation or a draw - the players are high-skilled enough to know which.

Because I am still a low-level player, I perform the traditional resignation by turning over my king and immediately offer a handshake so there is no confusion that I resign.  Verbally saying, "I resign" is just as effective.

I just say “I resign.”

I’ve always thought just ignoring my opponent after he/she offers a draw could be seen as rude. Grandmasters tend to sort of shrug and hold out their hand palm up, but I haven’t seen them after declining a draw offer. I just wondered if there is a way to decline with good sportsmanship.

I definitely understand wanting to avoid having the appearance of being rude.  But hopefully your opponent understands if you ignore them; after all, you may need some time to think if you have actually reached a drawn position or not.

Yes, simply making a move without verbally replying is a tacit rejection of the draw offer.

you're not allowed to talk though

resign = king knock handshake after, draw equals handshake question face thingy maboby 

lol

FIDE rule 9.1 a. The rules of a competition may specify that players cannot agree to a draw,
whether in less than a specified number of moves or at all, without the consent of
the arbiter.
b. If the rules of a competition allow a draw agreement the following apply:
(1) A player wishing to offer a draw shall do so after having made a move on the
chessboard and before stopping his clock and starting the opponent’s clock.
An offer at any other time during play is still valid but Article 12.6 must be
considered. No conditions can be attached to the offer. In both cases the offer
cannot be withdrawn and remains valid until the opponent accepts it, rejects
it orally, rejects it by touching a piece with the intention of moving or
capturing it, or the game is concluded in some other way.

-------------

Says right there that the player can reject the draw "orally" (by which they must mean, "verbally").

Although verbal is often used to signify speech, as in "verbal abuse," in a literal sense, "verbally" means using words, either written or spoken, while "orally" only means spoken.  Thus "orally" is the unambiguous term.  In law, the term "oral contract" is used to signify a spoken agreement. The term "verbal contract" is to be avoided.

chamo 2071 wrote: "you're not allowed to talk, though"

There is no rule that says one is "not allowed to talk."  Under the FIDE Laws of Chess 11.5 "It is forbidden to distract or annoy the opponent in any manner whatsoever. This includes unreasonable claims, unreasonable offers of a draw or the introduction of a source of noise into the playing area. "

Obviously, distraction is in the eye of the beholder. In tournament play, speech is usually limited to terse phrases like, “Draw?” or “J’adoube. (I adjust)” 

In science when writing a paper or doing a thesis or doctorate we use "verbal" because "oral" just doesn't sound right....well you know?

You could use the Harry Beltic method and say "Hell no!" Not sure if you'll make new friends that way though...

At the professional level they just make a move

When I took my American Board of Surgery exams, we referred to the "qualifying exam" as the written exam and the certifying exam as the orals.  (Of course, the "written" exam wasn't really written, it was done on a computer, I guess a throwback to earlier times) As an example, here's some text from the American Board of Surgery web site: 

• Successful completion of the General Surgery Qualifying and Certifying Exams is required for board certification in general surgery. Typically, the QE is an 8-hour, multiple-choice exam held once per year on a single day at computer-testing centers across the U.S. The CE is an oral exam consisting of 3 consecutive 30-minute sessions. It is held 5 times per year in various U.S. cities.