The problem of communication over binary dirty paper (DP) using nested polar codes is considered. An improved scheme, focusing on low delay, short to moderate blocklength communication is proposed. Successive cancellation list (SCL) decoding with properly defined CRC is used for channel coding, and SCL encoding without CRC is used for source coding. The performance is compared to the best achievable rate of any coding scheme for binary DP using nested codes. A well known problem with nested polar codes for binary DP is the existence of frozen channel code bits that are not frozen in the source code. These bits need to be retransmitted in a second phase of the scheme, thus reducing transmission rate. We observe that the number of these bits is typically either zero or a small number, and provide an improved analysis, compared to that presented in the literature, on the size of this set and on its scaling with respect to the blocklength when the power constraint parameter is sufficiently large or the channel crossover probability sufficiently small.