I have a fixed number of people (resources) and they are working on several projects all year long, generally 4 or 5 at the same time.
In general, 2 hours spent on a project Monday, 4 on Thursday, one on the following Monday, and so on.
All resources do this. It's like that because of customer availability, vendor schedules, other responsibilities to tend to, and so on.
I'd like to get an overall picture of when someone is available. That is, I'd like to know when Joe Smith is free, booked, overloaded, and so on across all projects, and not just in reference to one particular project.
I'm not sure Open Project is the right tool for this, but if it is, how can I do that, and if it is not, are there any suggestions for a resource management solution that ties to projects?
I think you should try eResource Scheduler Software by Enbraun. It performs every single thing that you have mentioned. It gives you an overview of your resource's utilization, availability, over-bookings etc. It allows you to share real-time information so that you and your respective colleagues will remain on the same page always. You can track your resources as and when needed.
To evaluate its features you can have a free trial or you can ask for personal web demo in your environment.