This work treats the problem of error-resilient DNA searching via oblivious evaluation of finite automata, formulated as follows: a client has a DNA sequence, and a service provider has a pattern that corresponds to a genetic test. Error-resilient searching can be achieved by representing the pattern as a finite automata and evaluating it on the DNA sequence, which is treated as the input to the automaton, where privacy of both the pattern and the DNA sequence must be preserved. Interactive solutions to this problem already exist, but can be a burden on the participating parties. Thus, in this work we propose techniques for secure outsourcing of oblivious evaluation of finite automata to computational servers, such that the servers do not learn any information. Our techniques are applicable to any type of finite automata, but the optimizations are tailored to the setting of DNA searching (i.e., when the alphabet size is small, etc.).