Large language models (LLMs) excel on static benchmarks, but their performance across multi-turn conversations, which better reflect real-world usage, remains understudied. Addressing this gap is critical in high-stakes settings like healthcare, where patients and clinicians are turning to LLM chatbots to address their medical inquiries. Here, we introduce the "stick-or-switch" (SoS) framework, which partitions a question-answer space into multiple sequential presentations to model two safety-centric behaviors: conviction (i.e., sticking to a correct answer selection or abstention against incorrect suggestions) and flexibility (i.e., switching to a correct suggestion when it is introduced). Evaluating 17 LLMs across three clinical benchmarks, we observe a pervasive conversation tax, where partitioning an answer-space into sequential presentations reduces end-to-end accuracy and abstention against incorrect suggestions by an average of up to 30%, reaching 65% in certain models. We also observe blind switching, where models transition an initial abstention to incorrect and correct suggestions at near-identical rates reaching 50%. Finally, we show that increasing model scale mitigates some of these conversational inefficacies while exacerbating others, such as a higher propensity to adopt an incorrect suggestion from an initial abstention. Together our findings demonstrate that the general proficiency captured by static benchmarks do not translate over multi-turn dialogues.