Using the validation data I'm unclear on the exact process of using the validation data.

Let's say that I fit my neural network model and adjust hyperparameters using the training set and validation set. Do I then evaluate the test set on this model? Or do I recombine the validation and training sets and fit a fresh model with the hyperparameters that I found during the validation phase, and then evaluate on the test data? I have seen a number of different notebooks and examples that do both ways.

Surely, once I've found my hyperparameters, it makes sense to fit a fresh model using the full training set (recombined with validation set), since the validation loss has no effect on the weights.
The common procedure is this:

Fit the model on train split and adjust hyperparameters using the training split and validation split, as you already said.

Then, to get a "final measure" of performance, train model on both training and validation splits using the hyperparams found during validation phase, and evaluate on test split.