Pattern searches, a key operation in many data analytic applications, often deal with data represented by multiple states per dimension. However, hash tables, a common software-based pattern search approach, require a large amount of additional memory, and thus, are limited by the memory wall. A hardware-based solution is to use content-addressable memories (CAMs) that support fast associative searches in parallel. Ternary CAMs (TCAMs) support bit-wise Hamming distance (HD) based searches. Detecting the HD of vectors with multiple states per dimension (i.e., multi-state Hamming distance (MSHD)) can be implemented on TCAMs with one-hot encoding, but requires one TCAM cell per state, leading to a higher area, latency, and energy overhead. We propose a Ferroelectric FET (FeFET)-based multi-state CAM design, MHCAM, which implements MSHD searches in a dense FeFET-based memory array. MHCAM only uses $\lceil log_{2} s \rceil ~2$ FeFET CAM cells to represent $s$ states or symbols per dimension, and can be reconfigured to 2-bit/4-bit/6-bit/8-bit dimensions. A low-cost sensing circuit with matchline voltage scaling technique is introduced to perform both exact match and threshold match. We use DNA and protein pre-alignment filtering as application case studies to evaluate the application-level benefit of MHCAM. DNA and protein pre-alignment filtering achieve $3.8\times /4.7\times$ speedup and $1.7\times /1.8\times$ energy improvement compared with the state-of-the-art 2FeFET TCAM-based implementation.