Essentially the key is feedback, ideally instantaneous and non-disruptive. So one person reads a prompt and starts their response. The other person starts the timer and merely raises their hand and/or counts the number of "undesirable" words/sounds that come out.
Focus on one skill at a time, e.g. if you are focusing on cutting out filler words, do not even think about eye contact. If you are focusing on eye contact, do not even think about the content of your words.
[0] https://www.speechskills.com/speechskills-resources/soundbit... - but these are really just a bunch of prompt cards. You can make your own or surely find some prompt set for much less, just make sure they're not intellectually or emotionally challenging. The whole point is just verbal processing!