Use a 4x6x16 top beam for a cantilevered swing set

Any 'credible and/or official sources' will tell you to hire a Structural Engineer, which is what I'll say also. There are too many variables and unknowns for Internet advice to suffice. Prepare well to minimize costs, supplying all dimensions and drawings, and schedule a site visit.

The Engineer may be able to give the details for a top truss run perpendicular to the existing joists. Gravity can't tell the difference between top and bottom. A top truss or beam will create a lump or ridge on the attic storage floor. The 2x8 joists would be held up with strap hangers. The two ends of the truss must be supported, which could be difficult if there's a two car garage door. You'd need to remove siding to slide the beam in place.

Or as @Ecnerwal wrote the engineer may prefer to retrofit the existing peaked roof into a series of trusses by adding web elements. These might go in very easily from the inside.

Family Handyman has an article that has a good analogous bottom support example. Prepare to DIY the work, not the engineering.

You may want to go with a TJI (truss joist) that is made by one of these companies (Weyerhaeuser, Georgia Pacific, etc.,). Unless you use a 2 lb foam in the rafter bays, you will need something 14" or taller to get enough insulation in the ceiling to meet code. Of coarse, this depends on the area you live in. Different regions have different R-value requirements.