@Travis24 - this thread is mainly dealing with projects in the “Projects” section of the sidebar (instead of starred projects). I don’t have any issues with moving around starred items (I use the browser version, I assume this holds true for the desktop app but not sure), so your best bet is probably to troubleshoot using these steps and then contact support if you can’t resolve.
RE: saved searches, they are technically not gone if you un-star (i.e., the direct URL will still take you to the search) but it’s definitely frustrating that there’s not a more persistent way to manage them. There aren’t too many related feedback requests (you could create one in the English Forum > Product Feedback category if you’d like), but here’s a sort of related request you could vote for: