We describe the first algorithms that satisfy the standard notion of node-differential privacy in the continual release setting (i.e., without an assumed promise on input streams). Previous work addresses node-private continual release by assuming an unenforced promise on the maximum degree in a graph, but leaves open whether such a bound can be verified or enforced privately. Our algorithms are accurate on sparse graphs, for several fundamental graph problems: counting edges, triangles, other subgraphs, and connected components; and releasing degree histograms. Our unconditionally private algorithms generally have optimal error, up to polylogarithmic factors and lower-order terms. We provide general transformations that take a base algorithm for the continual release setting, which need only be private for streams satisfying a promised degree bound, and produce an algorithm that is unconditionally private yet mimics the base algorithm when the stream meets the degree bound (and adds only linear overhead to the time and space complexity of the base algorithm). To do so, we design new projection algorithms for graph streams, based on the batch-model techniques of Day et al. 2016 and Blocki et al. 2013, which modify the stream to limit its degree. Our main technical innovation is to show that the projections are stable -- meaning that similar input graphs have similar projections -- when the input stream satisfies a privately testable safety condition. Our transformation then follows a novel online variant of the Propose-Test-Release framework (Dwork and Lei, 2009), privately testing the safety condition before releasing output at each step.