Oniun_'s journey

I recently interviewed for a position with the president and the director of operations at a company. On Friday, I received a call from the president offering me the job, welcoming me to the team, and informing me that my start date is in two weeks. The delay is due to the fact that both he and the director will be on vacation, and they want to be in-house when I start. While I’m excited about the offer, I have a few concerns. First, I worry that I might be forgotten by the time my start date arrives. Second, I noticed that the job posting is still live on Indeed, which makes me anxious that they might reconsider or look at other candidates. I also have no idea if they’ve contacted my previous employer yet, and I fear that could jeopardize my offer. I don’t believe a formal offer letter is coming; it seems I’ll just be signing documents on my first day in mid-August. I’m unsure if I should reach out to anyone with questions or to remind them that I’m still interested. I’m debating whether sending a casual email in a few days would be a good idea to check in and express my enthusiasm. Am I overthinking this? Should I go ahead and send that email?