Although computational linguistic methods—such as topic modelling, sentiment analysis and emotion detection—can provide social media researchers with insights into online public discourses, it is not inherent as to how these methods should be used, with a lack of transparent instructions on how to apply them in a critical way. There is a growing body of work focusing on the strengths and shortcomings of these methods. Through applying best practices for using these methods within the literature, we focus on setting expectations, presenting trajectories, examining with context and critically reflecting on the diachronic Twitter discourse of two case studies: the longitudinal discourse of the NHS Covid-19 digital contact-tracing app and the snapshot discourse of the Ofqual A Level grade calculation algorithm, both related to the UK. We identified difficulties in interpretation and potential application in all three of the approaches. Other shortcomings, such the detection of negation and sarcasm, were also found. We discuss the need for further transparency of these methods for diachronic social media researchers, including the potential for combining these approaches with qualitative ones—such as corpus linguistics and critical discourse analysis—in a more formal framework.