Hmm, I've never considered timing before. It's a possibility. Thanks, Bree123, for the link. I'd love to test the timing with the method in the link (and with quiltedsunshine's suggestion of sewing a zig zag), but I've now found that I'm unable to insert a needle at all.
Looked at the bar with a mirror from different angles, can't tell what's up there blocking it…
It does sound like I might need to get the machine serviced. Before I do that, I'll check the needle bar alignment just in case it's less time-consuming than sending the machine out.
Thanks, everyone! Been a long day, so I really appreciate these suggestions!