Software Engineering researchers are increasingly using Natural Language Processing (NLP) techniques to automate Software Vulnerabilities (SVs) assessment using the descriptions in public repositories. However, the existing NLP-based approaches suffer from concept drift. This problem is caused by a lack of proper treatment of new (out-of-vocabulary) terms for the evaluation of unseen SVs over time. To perform automated SVs assessment with concept drift using SVs' descriptions, we propose a systematic approach that combines both character and word features. The proposed approach is used to predict seven Vulnerability Characteristics (VCs). The optimal model of each VC is selected using our customized time-based cross-validation method from a list of eight NLP representations and six well-known Machine Learning models. We have used the proposed approach to conduct large-scale experiments on more than 100,000 SVs in the National Vulnerability Database (NVD). The results show that our approach can effectively tackle the concept drift issue of the SVs' descriptions reported from 2000 to 2018 in NVD even without retraining the model. In addition, our approach performs competitively compared to the existing word-only method. We also investigate how to build compact concept-drift-aware models with much fewer features and give some recommendations on the choice of classifiers and NLP representations for SVs assessment.