Manipulating Deformable Linear Objects (DLOs) is a challenging task for a robotic system due to their unpredictable configuration, high-dimensional state space and complex nonlinear dynamics. This paper presents a framework addressing the manipulation of DLOs, specifically targeting the model-based shape control task with the simultaneous online gradient-based estimation of model parameters. In the proposed framework, a neural network is trained to mimic the DLO dynamics using the data generated with an analytical DLO model for a broad spectrum of its parameters. The neural network-based DLO model is conditioned on these parameters and employed in an online phase to perform the shape control task by estimating the optimal manipulative action through a gradient-based procedure. In parallel, gradient-based optimization is used to adapt the DLO model parameters to make the neural network-based model better capture the dynamics of the real-world DLO being manipulated and match the observed deformations. To assess its effectiveness, the framework is tested across a variety of DLOs, surfaces, and target shapes in a series of experiments. The results of these experiments demonstrate the validity and efficiency of the proposed methodology compared to existing methods. Project website at https://sites.google.com/view/dlo-manipulation.