Flaky tests exhibit non-deterministic behavior during execution and they may pass or fail without any changes to the program under test. Detecting and classifying these flaky tests is crucial for maintaining the robustness of automated test suites and ensuring the overall reliability and confidence in the testing. However, flaky test detection and classification is challenging due to the variability in test behavior, which can depend on environmental conditions and subtle code interactions. Large Language Models (LLMs) offer promising approaches to address this challenge, with fine-tuning and few-shot learning (FSL) emerging as viable techniques. With enough data fine-tuning a pre-trained LLM can achieve high accuracy, making it suitable for organizations with more resources. Alternatively, we introduce FlakyXbert, an FSL approach that employs a Siamese network architecture to train efficiently with limited data. To understand the performance and cost differences between these two methods, we compare fine-tuning on larger datasets with FSL in scenarios restricted by smaller datasets. Our evaluation involves two existing flaky test datasets, FlakyCat and IDoFT. Our results suggest that while fine-tuning can achieve high accuracy, FSL provides a cost-effective approach with competitive accuracy, which is especially beneficial for organizations or projects with limited historical data available for training. These findings underscore the viability of both fine-tuning and FSL in flaky test detection and classification with each suited to different organizational needs and resource availability.