Recently, rate- $1/n$ zero-terminated (ZT) and tail-biting (TB) convolutional codes (CCs) with cyclic redundancy check (CRC)-aided list decoding have been shown to closely approach the random-coding union (RCU) bound for short blocklengths. This paper designs CRC polynomials for rate- $(n-1)/n$ ZT and TB CCs with short blocklengths. This paper considers both standard rate- $(n-1)/n$ CC polynomials and rate- $(n-1)/n$ designs resulting from puncturing a rate- $1/2$ code. The CRC polynomials are chosen to maximize the minimum distance $d_{\min }$ and minimize the number of nearest neighbors $A_{d_{\min }}$ . For the standard rate- $(n-1)/n$ codes, utilization of the dual trellis proposed by Yamada et al. lowers the complexity of CRC-aided serial list Viterbi decoding (SLVD). CRC-aided SLVD of the TBCCs closely approaches the RCU bound at a blocklength of 128. This paper compares the FER performance (gap to the RCU bound) and complexity of the CRC-aided standard and punctured ZTCCs and TBCCs. This paper also explores the complexity-performance trade-off for three TBCC decoders: a single-trellis approach, a multi-trellis approach, and a modified single-trellis approach with pre-processing using the wrap around Viterbi algorithm.