For projection alone, sensor size is usually not the deciding factor. Both APS-C and full-frame cameras typically produce far more resolution than most TVs and projectors can display.

A 1080p display is about 2 megapixels, and even 4K is only about 8.3 megapixels. Many digital cameras exceed that easily, so either format can provide enough detail for projected viewing.

In practice, your projected image will be limited more by the resolution of the TV/projector than by whether the camera is APS-C or full-frame. Technique, composition, and overall image quality matter more here than sensor size.

If you want to keep using your existing slide projector, one option is to have slides made from your digital files. That may preserve more detail than a typical consumer digital projector can show.

So if your main goal is projecting images digitally, there’s no strong projection-specific advantage to full-frame over APS-C.