My shared links show no picture, or the wrong one

Switch on Add Open Graph and X card tags under Networks → Open Graph Tags. Then either give the post a featured image, or choose a Share picture in the post's Share buttons box, which is used instead of the featured image when a post is shared. For posts without either, set a Default share picture on the same panel; without one, the Site Icon is used, on the small card. The card in the Share buttons box shows which picture a share will carry before you save, in the block editor and the classic editor. If an SEO plugin is active, the panel and the box say so, and the picture is set in that plugin instead.

Facebook keeps its own copy of a preview. After a change, paste the address into Facebook's Sharing Debugger and press Scrape Again.