Neural networks (NNs) are increasingly used in always-on safety-critical applications deployed on hardware accelerators (NN-HAs) employing various memory technologies. Reliable continuous operation of NN is essential for safety-critical applications. During online operation, NNs are susceptible to single and multiple permanent and soft errors due to factors such as radiation, aging, and thermal effects. Explicit NN-HA testing methods cannot detect transient faults during inference, are unsuitable for always-on applications, and require extensive test vector generation and storage. Therefore, in this paper, we propose the \emph{uncertainty fingerprint} approach representing the online fault status of NN. Furthermore, we propose a dual head NN topology specifically designed to produce uncertainty fingerprints and the primary prediction of the NN in \emph{a single shot}. During the online operation, by matching the uncertainty fingerprint, we can concurrently self-test NNs with up to $100\%$ coverage with a low false positive rate while maintaining a similar performance of the primary task. Compared to existing works, memory overhead is reduced by up to $243.7$ MB, multiply and accumulate (MAC) operation is reduced by up to $10000\times$, and false-positive rates are reduced by up to $89\%$.