Over the past few years, researchers have been focusing on the identification of offensive language on social networks. In places where English is not the primary language, social media users tend to post/comment using a code-mixed form of text. This poses various hitches in identifying offensive texts, and when combined with the limited resources available for languages such as Tamil, the task becomes considerably more challenging. This study undertakes multiple tests in order to detect potentially offensive texts in YouTube comments, made available through the HASOC-Offensive Language Identification track in Dravidian Code-Mix FIRE 2021.11https://competitions.codalab.org/competitions/31146. To detect the offensive texts, models based on traditional machine learning techniques, namely Bernoulli Naïve Bayes, Support Vector Machine, Logistic Regression, and K-Nearest Neighbor, were created. In addition, pre-trained multilingual transformer-based natural language processing models such as mBERT, MuRIL (Base and Large), and XLM-RoBERTa (Base and Large) were also attempted. These models were used as fine-tuner and adapter transformers. In essence, adapters and fine-tuners accomplish the same goal, but adapters function by adding layers to the main pre-trained model and freezing their weights. This study shows that transformer-based models outperform machine learning approaches. Furthermore, in low-resource languages such as Tamil, adapter-based techniques surpass fine-tuned models in terms of both time and efficiency. Of all the adapter-based approaches, XLM-RoBERTa (Large) was found to have the highest accuracy of 88.5%. The study also demonstrates that, compared to fine-tuning the models, the adapter models require training of a fewer parameters. In addition, the tests revealed that the proposed models performed notably well against a cross-domain data set. •Identifying the predictive features that distinguish offensive texts in Tamil.•Measuring the efficacy of transformer models in classifying offensive texts in Tamil.•Testing the cross-domain ability of the proposed models on the misogynous texts.