For best quality, the community consensus is that many consumer slide scanners—especially flatbeds with transparency adapters—are disappointing for 35mm slides. Slides have dense shadows and high contrast, so they need a scanner with strong dynamic range and good illumination.

Good options mentioned:

• A dedicated film scanner is preferred over a flatbed.
• Older Nikon Coolscan models are well regarded, though discontinued and typically found used.
• If the slides are Kodachrome, the Nikon Super Coolscan 9000 ED is specifically noted as a strong option because its dust/scratch removal system works better with Kodachrome than many other scanners.

Alternatives:

• A good photo lab may give the best results, especially for valuable family archives.
• A digital camera with a macro lens can also be a viable digitizing setup.

About your projector test: the “lights too light and darks too dark” issue is basically limited dynamic range / excessive contrast, which is why projection-copying usually performs poorly compared with proper scanning or camera digitizing.