While Automatic Speech Recognition (ASR) systems have shown impressive performance in languages having sufficient annotated speech data like English, their performance is still limited for low-resource, dialect rich languages like Tamil. Tamil poses further challenges because of its extremely high regional variation in dialects that manifest in varying vocabulary, pronunciations, and even syntactic structures. To address these challenges, we present a unified framework WhisTam based on the Whisper medium model, which performs speech transcription and dialect classification jointly within a single system. Our method is evaluated against speech samples from four regional dialects and achieves a macro F1-score of 0.53 and a Word Error Rate (WER) of 0.55 for dialect classification and transcription respectively, ranking 2nd in the dialect classification task and 3rd in the transcription task in the DravidianLangTech@ACL 2026 shared task on Dialect-based Speech Recognition and Classification in Tamil. These findings emphasize the challenges in dialectal Tamil ASR as well as the promise of multi-task learning for low-resource languages. Our implementation is publicly available at: https://github.com/rwd51/DravidianLangTech-Wave2Word.