The Early Fault-Tolerant (EFT) era is emerging, where modest Quantum Error Correction (QEC) can enable quantum utility before full-scale fault tolerance. Quantum optimization is a leading candidate for early applications, but protecting these workloads is critical since they will run on expensive cloud services where providers could learn sensitive problem details. Experience with classical computing systems has shown that treating security as an afterthought can lead to significant vulnerabilities. Thus, we must address the security implications of quantum computing before widespread adoption. However, current Secure Quantum Computing (SQC) approaches, although theoretically promising, are impractical in the EFT era: blind quantum computing requires large-scale quantum networks, and quantum homomorphic encryption depends on full QEC. We propose application-specific SQC, a principle that applies obfuscation at the application layer to enable practical deployment while remaining agnostic to algorithms, computing models, and hardware architectures. We present Enigma, the first realization of this principle for quantum optimization. Enigma integrates three complementary obfuscations: ValueGuard scrambles coefficients, StructureCamouflage inserts decoys, and TopologyTrimmer prunes variables. These techniques guarantee recovery of original solutions, and their stochastic nature resists repository-matching attacks. Evaluated against seven state-of-the-art AI models across five representative graph families, even combined adversaries, under a conservatively strong attacker model, identify the correct problem within their top five guesses in only 4.4% of cases. The protections come at the cost of problem size and T-gate counts increasing by averages of 1.07x and 1.13x, respectively, with both obfuscation and decoding completing within seconds for large-scale problems.