Text understanding often suffers from domain shifts. To handle testing domains, domain adaptation (DA) is trained to adapt to a fixed and observed testing domain; a more challenging paradigm, test-time adaptation (TTA), cannot access the testing domain during training and online adapts to the testing samples during testing, where the samples are from a fixed domain. We aim to explore a more practical and underexplored scenario, continual test-time adaptation (CTTA) for text understanding, which involves a sequence of testing (unobserved) domains in testing. Current CTTA methods struggle in reducing error accumulation over domains and enhancing generalization to handle unobserved domains: 1) Noise-filtering reduces accumulated errors but discards useful information, and 2) accumulating historical domains enhances generalization, but it is hard to achieve adaptive accumulation. In this paper, we propose a CTTA-T (continual test-time adaptation for text understanding) framework adaptable to evolving target domains: it adopts a teacher-student framework, where the teacher is domain-aware and generalized for evolving domains. To improve teacher predictions, we propose a refine-then-filter based on dropout-driven consistency, which calibrates predictions and removes unreliable guidance. For the adaptation-generalization trade-off, we construct a domain-aware teacher by dynamically accumulating cross-domain semantics via incremental PCA, which continuously tracks domain shifts. Experiments show CTTA-T excels baselines.