Context: Test case generation is a critical yet challenging task in software development. Recently, AthenaTest – a Deep Learning (DL) approach for generating unit test cases has been proposed. However, our revisiting study reveals that AthenaTest can generate less than one-fifth of the test cases correctly, due to a lack of assertion knowledge and test signature verification. Objective: This paper introduces A3Test, a novel DL-based approach to the generation of test cases, enhanced with assertion knowledge and a mechanism to verify consistency of the name and signatures of the tests. A3Test aims to adapt domain knowledge from assertion generation to test case generation. Method: A3Test employs domain adaptation principles and introduces a verification approach to name consistency and test signatures. We evaluate its effectiveness using 5,278 focal methods from the Defects4j dataset. Results: Our findings indicate that A3Test outperforms AthenaTest and ChatUniTest. A3Test generates 2.16% to 395.43% more correct test cases, achieves 2.17% to 34.29% higher method coverage, and 25.64% higher line coverage. A3Test achieves 2.13% to 12.20% higher branch coverage, 2.22% to 12.20% higher mutation scores, and 2.44% to 55.56% more correct assertions compared to both ChatUniTest and AthenaTest respectively for one iteration. When generating multiple test cases per method A3Test still shows improvements and comparable efficacy to ChatUnitTest. A survey of developers reveals that the majority of the participants 70.51% agree that test cases generated by A3Test are more readable than those generated by EvoSuite. Conclusions: A3Test significantly enhances test case generation through its incorporation of assertion knowledge and test signature verification, contributing to the generation of correct test cases.