Can I assign a project only to the internal funding source?

Yes, for awards that are enabled for cost sharing, you can assign a project only to an internal funding source.

If you select a project or project template that's associated with a billable project type, but the project is assigned only to an internal funding source, then the cost transactions for the internal funding source are set to nonbillable. Also, projects that are assigned to an internal funding source aren't assigned to a contract.