I would like the priority and severity field of my project to only be editable by myself or managers in my team. Is there a way to do this? Also is there a way to do this where it does not globally affect other projects?
In principle, no. Jira does not do field level security (because hiding bits of issues leads to confusion and doubt)
You can do the "only set by" thing though. Edit the workflow, creating transitions that are protected by "manager only" conditions that then go through screens that include the protected fields. Then remove those fields from the "edit" screen so they can only be changed via the (protected) transitions.
You could set up issue security for your project and set a security level but, that would make an issue inaccessible for anyone not in the security level.
You could also look at script runner behaviors to lock down the fields to only you and the other managers.