If, for example, someone said : "How are you today?", how would you go about "translating" this into sounding like they were talking with a full-mouth? Like, phonetically.
For me, phonetically translating mouth-full speaking would deter my focus and make me dwell on words missed and other incomprehensible text which would arise.
Instead, I would focus on the experience of the interaction:
There the three of us sat talking about the business proposal. Sam was focused in reading the details while Eric continued eating and interrupting with questions.
Eric asks "So, what are the (munch) proposed (chew) percentage distributions?"
Sam responds that she hasn't made it to that section yet all the while Eric gulps his drink and tells her to read faster. He also muttered something else but I was too grossed out by the sputter of potatoes and gravy which was exiting from his mouth.
But, if you are truly set on phonetically spelling mouth-full words then you can write the dialogue and read it out loud while eating and record yourself. This will provide the most realistic result.
However, you might need to provide context to avoid confusing the reader:
As I entered their office, they greeted me with "Hi, mmph, ghow are you?"
As I entered their office in the middle of lunch hour, they greeted me with "Hi, mmph, ghow are you?"
The former makes you question:
- Did the person have a stroke upon greeting?
- Did the person hold back a sneeze?
- Were they totally caught off-guard at an awkward time?
- Did they hit their funny bone?
Although I don't recommend trying to reproduce the phonetic sound; it is pretty easy to figure out. The tongue cannot touch the palate (roof of the mouth) or the teeth because it is blocked by food; so plosives like "T", "K", palate "G" (like in "Get"), "D" are suppressed. Lip plosives ("P", "B") could still be voiced.
Replace the plosives that cannot be voiced with single apostrophes.
"That's better" would become "'a's be'er". A Word like "dog" that begins and ends with a plosive would be unintelligible; "'o'".
It depends on what sort of speech you tend to write. Personally, I would approach this like method acting. Write the character's script, make a sandwich, turn on a voice recorder, then read the script with your mouth full.
When listening to the playback, write down what you hear. If it is meant to be unintelligible, don't try to shoehorn your original script in there to make words guessable. Have the narrator indicate why there is confusion and (probably) have the other characters ask for clarification (depending on the power dynamic of the group).
I walked up to her and she greeted me with a "ghffuh mmnet". "What?" came my reflexive response. After she swallowed the bite of what looked to be a deliciously ripe tomato, she clarified herself: "'Just a minute', I said."
The convention to represent any deviation from normal speech in literary writing is by naming the deviation in the accompanying description.
- "Wait a minute, I'll just chew and swallow," John mumbled with his mouth full.
- Joan couldn't make out what Heather was saying. She had her mouth full and all that came out were strangled noises and some bread crumbs.
- "That's it!" Bob shouted / whispered / said with his Southern drawl / Russian accent etc.
Loud speech isn't printed in bold face, whispered speech isn't printed in smaller letters, you do not (commonly) write in dialect (unless you want to limit your audience to those who understand it), and so on. The convention is to write Standard English and to tell the reader in which way the speaker doesn't speak it.
Of course, many writers have attempted to write what they think is "ye olde English" or some other variant of the language, mostly with the unintended effect of irritating their readers. I recommend not to attempt it.