Your Capacity Planning and Prioritization Processes Need Re-Engineering
You appear to be attempting to solve the wrong problem by treating this as an individual-availability issue. This is most likely because you have a weirdly-matrixed organization that isn't based around project teams or team capacity planning, but apparently based on the notoriously unsuccessful "Is Joe busy right now?" methodology.
Your team of eight should have a variable but generally predictable capacity. In Scrum, this is your velocity range over some trailing historical period. Whether or not you follow Scrum, this sort of historical estimate of future capacity is often your most accurate guide to determining whether a team may have capacity for additional work.
In addition, each project team should not be expected to context-switch without a visible and costly penalty to productivity. Your organization is clearly missing both a Product Owner for each project and a scaled framework that can prioritize work across multiple projects. As a result, everyone treats their personal priorities as the priorities, and when everything is a "top priority" then nothing gets done.
Your organizational process clearly needs work. Consider Scrum, Kanban, and the available scaled frameworks like SAFe or a Scrum-of-Scrums. There's no silver bullet, but almost any formal framework will be better than what the organization is doing now.