I don’t know exactly the interface behind the registrar and the root zone operators; whether they send deltas or daily complete updates. But they should know the process and have done the right thing.
Likewise, Network Solutions should have imported the previous DS records to their intrerface as you transferred the zone in. Presumably they also transferred in the NS records, but maybe you had to supply those?
DNS is tricky to get right; DNSSEC even more so.