This paper introduces the development of an anthropomorphic soft robotic hand integrated with multiple flexible force sensors in the fingers. By leveraging on the integrated force sensing mechanism, grip state estimation networks have been developed. The robotic hand was tasked to hold the given object on the table for 1.5 s and lift it up within 1 s. The object manipulation experiment of grasping and lifting the given objects were conducted with various pneumatic pressure (50, 80, and 120 kPa). Learning networks were developed to estimate occurrence of object instability and slippage due to acceleration of the robot or insufficient grasp strength. Hence the grip state estimation network can potentially feedback object stability status to the pneumatic control system. This would allow the pneumatic system to use suitable pneumatic pressure to efficiently handle different objects, i.e., lower pneumatic pressure (50 kPa) for lightweight objects which do not require high grasping strength. The learning process of the soft hand is made challenging by curating a diverse selection of daily objects, some of which displays dynamic change in shape upon grasping. To address the cost of collecting extensive training datasets, we adopted one-shot learning (OSL) technique with a long short-term memory (LSTM) recurrent neural network. OSL aims to allow the networks to learn based on limited training data. It also promotes the scalability of the network to accommodate more grasping objects in the future. Three types of LSTM-based networks have been developed and their performance has been evaluated in this study. Among the three LSTM networks, triplet network achieved overall stability estimation accuracy at 89.96%, followed by LSTM network with 88.00% and Siamese LSTM network with 85.16%.