Is it appropriate to call a research professor by phone or to email one of his postdoctoral assistants if he doesn't reply to emails? I've just completed the first year of a biological sciences PhD program and I still haven't chosen a lab for my doctoral studies since 3 of the labs I rotated in were not good fits and another had insufficient funding. I recently reached out to another professor and on May 13th we had a Zoom meeting wherein he seemed very interested in me and said he would gladly let me rotate in his lab this summer and I would be welcome to do my doctoral studies if it turned out to be a good fit. Before we wrapped up our conversation, he said he would email me some scientific papers related to his research for the coming weekend and reach out to one of his postdoctoral assistants whom I would be working with to arrange a meeting time, presumably the following week.
However, a week has passed since our Zoom meeting and I haven't yet heard anything from him or the lab assistant he said he would contact. On the 16th I emailed him a friendly reminder about the papers he said he would send to me and on the 18th I asked if he had reached out to the lab assistant to arrange a meeting time. But as of this writing on the 21st, he still hasn't replied to me. If he doesn't respond by the 24th, I plan on sending another friendly email reminder. I'm very and anxious and unsure what to make of the professor's lack of response.
It's important to mention that this professor is the PI of a lab and head of a University department so I know he must be an extremely busy person with a lot of responsibilities. I should also note that before our Zoom meeting, he often required email reminders and one time did not respond at all to an email. In fact, he even forgot about the Zoom meeting we had on the 13th at first and had to reschedule for the following hour before we spoke.
If he doesn't reply to the reminder that I plan to send him on the 24th, I was wondering if it would be appropriate to try calling his office phone. I'm also considering emailing the postdoctoral lab assistant. During the Zoom meeting I asked if he wanted me to reach out to this assistant but he told me that he would do it himself because the assistant doesn't yet know me. I'm concerned that calling his office phone might come across as stalking or that reaching out to his lab assistant may violate the preference he expressed to contact him himself at the Zoom meeting. Would doing these things become appropriate after two weeks have passed since our last contact?
I'm generally a very anxious person by nature but this situation is significantly exacerbating it since my standing in the PhD program depends on me finding a lab by Fall. My ultimate fear is that his lack of replies in the past week means that he's no longer interested in having me in his lab but I don't know if this fear is well-founded or if I'm just being overly anxious. So, in summary, what do you think I should make of his lack of response in the past week and should I reach out to his assistant or call his office phone if he doesn't respond by next week?
Thank you so much!
3 of the labs I rotated in were not good fits and another had insufficient funding

Definitely be in contact with your department about this. They need to be aware you're having trouble finding a lab; doing 4 rotations without a home is a concerning circumstance, though I've known people in that same situation that found a lab and did quite well.

a week has passed

A week is almost nothing in academic time, especially for a busy professor and especially around this time of year when for many universities the semester has just ended and people are either finishing up with end-of-semester business (grades, graduations) or perhaps taking some time off after their semester responsibilities have ended.