Twang and Knödel both feature a narrowed epiglottic funnel via epiglottic retroversion, which, as people who treat swallowing disorders can attest to, is inextricable from tongue root motion. It is my assessment that the hyoglossus is not a likely candidate for the muscle responsible here, but I am given to understand that the aryepiglottic folds were recently found to actually attach to the palatopharyngeus rather than the epiglottis as was originally thought. Even so, it seems likely that this muscle is responsible as a certain tilting of the arytenoid cartilages is also observed during twang qualities. Yet if twang and Knödel both feature the retracted tongue and retroverted epiglottis, what might be the biomechanical difference that can account for the difference in sound? I believe it is a matter of whether the oropharyngeal isthmus is narrowed via the palatoglossus muscle, which would account not only for why many people have easier time finding nasal twang than oral twang, but also for why many will describe twang as nasal sounding unlike Knödel: The nasal quality is perhaps not primarily the nasal formants (which are weak since the nasal cavity is an inefficient resonator) but the bright quality resulting from the narrowed oropharyngeal isthmus. I believe the same movement of epiglottic retroversion will result in either twang or Knödel depending on whether the oropharyngeal isthmus is narrowed.
I mentioned that the aryepiglottic muscle is likely responsible for the epiglottic retroversion. Since this also draws the arytenoid cartilages forward, it leads to some anteroposterior narrowing at the vocal fold level, which would indeed change the vibration of the vocal folds as you suggest - namely the vertical phase difference would be increased. I would also note however that people often underestimate the amount of source-filter interaction and that changing the resonances of the vocal tract will tend to change the vibration of the vocal folds as well. In addition, the narrowed epiglottic funnel would increase this nonlinearity (Titze, 2008) and it would also result in changes to the supraglottic flow due to changes in the aerodynamics (eg. the Bernoulli effect)