Yes they do (the vocal chords can certainly be overdriven into producing distortion products not unlike the effect of amplifier clipping - and for basically the same reason) when shouting at the top of your voice.
It isn't just the greater volume that helps the 'shouted word' carry for greater distances and or cut through high ambient noise levels, it's also the much higher level of average energy in such speech compared to conversational levels of speaking.
An effect that didn't go unnoticed by the communist bloc countries' radio broadcasting propaganda divisions who developed a speech processor to mimic the effect without asking the radio news readers to do anything more than read out "The News" in a normal voice. The effect became famously known as "The Radio Bulgaria Effect". Anyone who has ever heard any of these "Shouty" stations on the shortwave broadcast bands will recognise the effect immediately.
For anyone wishing to mimic this effect with a simple audio amplifier (a line out signal driving a couple of anti-parallel Si diodes across a volume pot via a limiting resistor so that the resulting 1.4v Pk2Pk clipped audio can be attenuated to a user adjustable level) the resulting audio effect will be marred by the in-band IM distortion products that were obviated by properly designed speech compressors (whether by analogue clipping of an SSBSC translation of the audio to a few hundred KHz or by brute force digital signal processing techniques).
A reasonably effective speech compressor can be created using the basic audio signal clipping method when bandpass shaping of the audio is applied prior to the clipping stage to reduce IM by the lower bass frequencies with a similar if not entirely compensating inverse bandpass shaping of the output from the clipping stage.
Ideally, you'd want to split the speech band of 300 to 3200Hz into 4 or more bands, clipping each one independently before recombining them again but once you get to this stage of complexity, you might as well translate the audio up to 455KHz as a SSBSC signal and apply the clipping there where all the IM products will be removed several hundred KHz away from the bandpass filter used to feed the SSB demodulator which translates the resulting mess back down to its original audio frequency range, free of those troublesome IM distortion products but cursed now only by the less disruptive but inescapable harmonic distortions (or you can programme a DSP to create the equivalent effect without all the bulk of even a 'modern' "RF audio processor" with its need for crystal lattice filters, an oscillator driving a balanced modulator and demodulator along with ancillary audio buffering components built into a metal screening box.)
That "Radio Bulgaria Effect" might sound like gross distortion (and it certainly is) but it's a specially crafted form of distortion designed to enhance communication (of propaganda messages in this case) via a very noisy and unreliable channel, plagued by QRN and QRM in the shortwave broadcast bands. Whilst gross distortion due to overloading or overdriving an amplifier can, at fist hearing, be mistaken for the "Radio Bulgaria Effect", it falls woefully short of the intelligibility enhancing effect of the latter when dealing with a marginally usable communication channel.