I think it has already been confirmed that shards can be broken further into further shards and these minor shards form an intent of their own. There are enough examples of that in elantris, warbreaker and stormlight archives.
I think that when a vessel dies the shard simply passes to the next vessel with the highest connection. If there are two or more vessels, I do not think that the shard would break into minor shards based on the connections of the vessel. I think the shattering has to be deliberate and has to be done by an external entity, probably another vessel. This would mean that Adonalsium was probably 'killed' by the original sixteen vessels.