The weights stay fixed. The state does not.

Earlier, we separated training from inference:

Training changes the learned parameters. Inference uses them.

The same rule applies to an RNN. During inference, its learned weights stay fixed.

The hidden state is different. It is temporary information about the sentence being processed, so each new word changes it.

Training also runs these hidden-state updates so the RNN can make a prediction. Then the error from that prediction can change the learned weights.