Over recent decades, research in Artificial Intelligence (AI) has developed a broad range of approaches and methods that can be utilized or adapted to address complex optimization problems. As real-world problems get increasingly complicated, this requires an effective optimization method. Various meta-heuristic algorithms have been developed and applied in the optimization domain. This paper used and ameliorated a promising meta-heuristic approach named Crow Search Algorithm (CSA) to address numerical optimization problems. Although CSA can efficiently optimize many problems, it needs more searchability and early convergence. Its positioning updating process was improved by supporting two adaptive parameters: flight length (fl) and awareness probability (AP) to tackle these curbs. This is to manage the exploration and exploitation conducts of CSA in the search space. This process takes advantage of the randomization of crows in CSA and the adoption of well-known growth functions. These functions were recognized as exponential, power, and S-shaped functions to develop three different improved versions of CSA, referred to as Exponential CSA (ECSA), Power CSA (PCSA), and S-shaped CSA (SCSA). In each of these variants, two different functions were used to amend the values of fl and AP. A new dominant parameter was added to the positioning updating process of these algorithms to enhance exploration and exploitation behaviors further. The reliability of the proposed algorithms was evaluated on 67 benchmark functions, and their performance was quantified using relevant assessment criteria. The functionality of these algorithms was illustrated by tackling four engineering design problems. A comparative study was made to explore the efficacy of the proposed algorithms over the standard one and other methods. Overall results showed that ECSA, PCSA, and SCSA have convincing merits with superior performance compared to the others.