San Jose has Askarov. He is unproven yet at the NHL level, but he has been very good in the AHL and among prospects, he's one of the best. He might struggle a bit the next couple years, especially if the defense is hanging him out to dry, but he should be a quality backstop for a long time.
Defensively, you're right. They don't seem to have anyone of significance outside of Dickinson. I don't know much about the rest of their defensive prospects. There are a couple who were drafted in the 2nd round, but none are names I recognize.
The UFA market is weak, but they have the assets and the cap space to make trades. They can take on vets that other teams are looking to dump. They won't be long-term solutions, but they can potentially improve the team a bit until they can find better options.
If they end up drafting Schaefer, he could help them out considerably. They also have Dallas' 1st this year, Edmonton's 1st next year, and 4 2nd rounds picks. They might be able to snag another good dman or two with those picks. Of course, it will take time for those players to develop, but their forward group is still very young and developing as well.
Three years from now, they could be a very strong team with the right moves.