Mine did not, or seemed not to, re-seed. (I also often got two blooms per year.
But slugs instantly mowed down EVERY Delphinium seedling they could reach, so I think we might need to start them indoors (or isolated from slugs) until they can grow new leaves faster than slugs can eat them.
I also saw that once a plant was more than 18-20 inches high, the slugs lost almost all interest. As if they only like Delphinium "micro-greens".
Delphs HAVE TO be the easiest plants to collect and clean seeds from. I just waited until they were ripe, dry and rattling in their little cups. Then I leaned each flower stalk over and just POURED all the seeds into a big cardboard box (a trash bag or opened newspaper would work, too). Maybe I hand-picked one or two pieces of chaff out of the box, but then the seeds were 100% clean.