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 to 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 non-billable. Also, projects that are assigned to an internal funding source aren't assigned to a contract.